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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION.

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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  Updated on: 24 Jan 2017, 20:46
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Day-1



MOCK TEST-1


Conceptual Quiz on Divisibility and Prime numbers.

Number of Questions-> 100.

Sources-> The following questions are from various sources and quite a few are self made.


1)If z is divisible by all the integers from 1 to 5 inclusive,what is the smallest value of z?
Show Spoiler
60.


2)Data Sufficiency-> Is the positive integer z divisible by 12?
A)z is divisible by 3.
B)z is not divisible by 2.

Show Spoiler
B.


3)Data Sufficiency-> Is integer z divisible by 21?
A)z is divisible by 7.
B)z is divisible by 3.

Show Spoiler
C.


4)Data Sufficiency-> Is integer z divisible by 48?
A)z is divisible by 12.
B)z is divisible by 4.

Show Spoiler
E.


5)Data Sufficiency-> Is integer z divisible by 12?
A)z is divisible by 8.
B)z is divisible by 6.

Show Spoiler
C.


6)Data Sufficiency-> Is integer z divisible by 24?
A)z is divisible by 6.
B)z is divisible by 4.

Show Spoiler
E.


7)How many prime numbers between 1 and 100 are factors of 168?

Show Spoiler
Three.


8)How many Prime factors does 2180 have?

Show Spoiler
Three.
2180=2^2*5*109
Notice 109 is a prime number.
Hence, 2180 has three prime factors


9)What is the greatest prime factor of \(49^{19}-7^{35}\)
A)2
B)3
C)7
D)13
E)19
Show Spoiler
E.
49^19=7^38
Thus => 49^19-7^35=7^38-7^35 => 7^35(7^3-1) = 7^35*342=> 7^35*2*3^2*19
Hence greater Prime factor=19.


10)If n is the product of integers from 1 to 12 exclusive,how many prime factors does n have?

Show Spoiler
Five.
Product = \(2^8*3^4*5^2*7*11\).


11)If 2,3,5 are the prime factors of N,what is the value of N?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
Show Spoiler
E.
N=\(2^a*3^b*5^c* P^d\)
Where p is some other prime number.
Basically,There are ∞ numbers with the same set of primes.


12)If 2,3,5 are the "ONLY" prime factors of K,what is the value of K?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
Show Spoiler
E.
K\(=2^a*3^b*5^c\)
Again -> There are ∞ numbers with the same set of primes.


13)Data Sufficiency -> What is the value of positive integer K?
A)2,5,7 are prime factors of K.
B)K<100
Show Spoiler
C.
From statement 1-> K = \(2^a*5^b*7^c*P^z\) ; for any prime number P.
There exist ∞ numbers with the same set of prime numbers.
Hence not sufficient.
From statement 2-> K<100. K can be any integer <100. Clearly not sufficient
Combing them -> Only Value of K possible is 70.


14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.

Show Spoiler
E.
Notice that the general expression for p will be --> 2 * Prime^even
E.g => p=2
or p=2*5^2
Etc.


15)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is an even number less than 15.
Show Spoiler
E.
From statement 1-> p=2 or 2*3^2 or 2*2^2 or 2* Any-prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=2 or p=2*2^2=8
Hence p can be 2 or 8
Thus not sufficient

Hence E.


16)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is an even number less than 15.
Show Spoiler
C.
From statement 1-> p=3 or 3*3^2 or 3*7^2 or 3*prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=3*2^2 =12 is the only possible value.
Hence Sufficient.
Hence C


17)10^25 – 560 is divisible by all of the following EXCEPT:-
A)11
B)8
C)5
D)4
E)3
Show Spoiler
E.
Clearly it is divisible by 8,5,4
The real Test here is choosing between 3 and 11.
10^t for any positive integer t will always leave a remainder 1 with 3.
e.g-> 10=> 3*3+1
100=3*33+1
1000=3*333+1
10000=3*3333+1

Hence 10^25=3m+1
Also 560=3k+2 (sum of digits =11,hence remainder with 3-> 2)
Thus 3m+1-3k-2 => 3x-1 for some integer x
Clearly it will never be divisible by 3.

Alternatively,we can also use the "Binomial Expansion".


18)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is a factor of 75

Show Spoiler
E.
Combing the two statements -> p can be 3 or 3*5^2



19)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is a factor of 60.

Show Spoiler
C.
Notice that by combing the two statements -> The only value of p possible is 2.


20)Data sufficiency-> If a and b are prime numbers,what is the value of a*b?
A)a-b=1
B)a=3

Show Spoiler
A.
Notice that the only consecutive numbers that are prime are -> 2,3
Hence From statement 1-> Product a*b=3*2=6


21)If x and y are prime numbers, each greater than 2, which of the following must be true?

I. x+y is an even integer

II. xy is an odd integer

III. (x/y) is not an integer

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Show Spoiler
B.
As x and y are prime numbers both greater than 2 => they both must be odd
Hence x+y=odd+odd=even
x*y=odd*odd=odd
Notice that if x=y => x/y will be 1. Hence III is not always true.
If x and y would happen to be different primes then III will be always true.


22)Which of the following statements must be true->
A)zero is neither prime nor composite.
B)one is neither prime nor composite.
C)-3 and 3 are both primes.
Show Spoiler
A and B.
Notice -3 is negative and hence cannot be prime


23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.

Show Spoiler
All statements are true.


24)Data Sufficiency->If p is a prime number,what is the value of p?
A)11<p<17
B)p is divisible by 13.

Show Spoiler
D.


25)Data Sufficiency->If A and B are positive integers, is A/B an integer?

(1) Every factor of B is also a factor of A.
(2) Every prime factor of B is also a prime factor of A.

Show Spoiler
A.
Statement 1->Using the property => "Every number is a factor of itself "
B must be a factor of B
Hence B must be a factor of A.
Thus A/B must be an integer -> Sufficient
Statement 2-> We can use test cases to Prove this insufficient
Case 1->\(\frac{2*3*7}{2*3*7}\) =1=Integer
Case 2->\(\frac{2*3*7*11}{2^5}\)=> Not an integer.
Hence insufficient
Hence A.
Source-> GMAT-Prep


26)Data Sufficiency->Is \(\frac{n}{14}\) an integer?

(1) \(n\) is divisible by 28.

(2) \(n\) is divisible by 70.
Show Spoiler
D.
Statement 1->n=28k = 14*2k=14k'
Statement 2->n=70k= 14*5k=14k'
Hence D.
Alternatively,we can use the property => "A number is divisible by factors as well as factors of its factors"


27)If x is a prime number, what is the value of x?

(1) There are a total of 50 prime numbers between 71 and x, inclusive.
(2) There is no integer n such that x is divisible by n and 1 < n < x.

Show Spoiler
A.
In statement 1->Notice that 71 is 20th prime.
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71.
Hence x will be 50th prime from 71.
Hence sufficient.
Statement 2-> This is just another way of saying that x is prime.
Hence not sufficient
Hence A.


28)Data Sufficiency->If p = (n)(5^x)(3^k), is p divisible by 10?

(1) n, x, and k are even.
(2) x > n > k > 0

Show Spoiler
C.
Firstly we are not given in p,x,k,n are an integers or not.
If p is an integer -> Then for p to be divisible by 10 => It must be dividable by 5 and 2=> It must have both 5 and 2 as its primes.
Hence => n must be even and n,k,x must be all pastime integers.
Statement 1->
All n,k,x are even.
Case 1-> \(-2*5^{-10}*3^{-10}\)=>p is not an integer
Case 2->\(2*5^2*3^4\)=> P is an integer and is dividable by 10.
Hence not sufficient.
Statement 2->
All are positive
Again lest use test cases
Case 1->\(1*5*3\)=> p is not divisible by 10
Case 2-> \(2*5*3\)=> p is divisible by 10
Combing the two statements => n must be a positive integer and x,k are positive even numbers.
Hence p must be dividable both 2 and 5.
Hence p must be dividable by 10.
Hence C.


29)Data Sufficiency->If x is a prime number,what is the value of x?

1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number

Show Spoiler
B.


30)If \(A\) is positive integer,then \(A\) , \(A^3\)and \(A^7\) will have the exact same prime factors.
A)True
B)False
Show Spoiler
True.
Property -> \(X\) and \(X^n\) always have the exact same prime factors.



31)Data Sufficiency->If y and x are positive integers, is y divisible by 3?

(1) y=x(x+1)
(2) y^2 is divisible by 9.
Show Spoiler
B.


32)Data Sufficiency->If y and x are positive integers, is y divisible by x?

(1) y = x^2 + x
(2) x has the same prime factors as y.
Show Spoiler
A.


33)Data Sufficiency->If z is a positive integer,is (z^31+7)^2 divisible by 4?
A)√z has five prime factors.
B)All prime factors of z^3 are greater than 7.

Show Spoiler
B.
Property in action -> For any number to be odd -> Two must not be its prime.
Also-> \(X\) and \(X^n\) always have the exact same prime factors.



34)If x is a positive odd integer and y is a negative even integer, which of the following must be true?

A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
Show Spoiler
C.


35)Data Sufficiency->How many positive prime numbers are less than the integer n?

(1) 14 < n < 20
(2) 13 < n < 17
Show Spoiler
B.
Remembering the first ten primes can be extremely helpful in these kind of questions.
2
3
5
7
11
13
17
19
23
29.


36)Data Sufficiency->If x and y are positive integers, is x divisible by y?

1) x is divisible by 5.
2) y is divisible by 5.
Show Spoiler
E.


37)Data Sufficiency->How many prime factors does \(x^{37}\) have?

(1) The number of prime factors of 16x is one more than the number of prime factors of \(x^4\)

(2) \(2x^{16}\) has 2 prime factors

Show Spoiler
C.This question is based on a very important property -> \(X\) and \(X^n\) always have the exact same prime factors.
So \(x^{37}\) will have the exact same prime factors as x.
Thus,here the question is asking us to get the number of prime factors of \(x\).
Statement 1-> \(16x\) has one more prime factor than \(x^4\)
Now \(x\) and \(x^4\) will have the exact same prime factors.
Hence \(16x\) has one more prime than \(x\)
\(16x=2^4*x\) => Hence 2 cannot be the prime factor of \(x\).
But we still don't have the number of prime factors of \(x\).
Hence not sufficient.
Statement 2-> \(2*x^{16}\) has two prime factors
Here again \(x^{16}\) will have the exact same primes as \(x\).
So as \(2*x^{16}\) has two prime factors => \(x\) can have either one prime factor if 2 isn't one of its prime .
Or two prime factors if 2 is one of its prime.
Hence not sufficient.
Combining the two statements=>
As 2 cannot be one of the primes of x => x must have only one prime factor
Hence sufficient.
Hence C.


38)If a positive integer x has 3 prime factors,how many prime factors does x^3 have?
A)27
B)6
C)3
D)1
E)cannot be determined
Show Spoiler
C.
\(X\) and \(X^n\) always have the exact same prime factors.


39)Data Sufficiency->If k is a positive integer, how many unique prime factors does 14k have ?

(1) k^4 is divisible by 100
(2) 50*k has 2 prime factors

Show Spoiler
C.
\(14k=2*7*k\)
To get the number of prime factors of 14k => we need the number of prime factors of k
Statement 1-> Using the property -> \(X\) and \(X^n\) always have the exact same primes => k must have both 2 and 5 as its primes.
General expression for k= \(2^a*5^b*P^z\)
Where a and b are non negative integers and P is any other prime number and z≥0
Hence not sufficient.
Statement 2->
Form this statement => k can be \(2^a\) or \(5^b\) or \(2^a*5^b\)
Or k=1
Hence not sufficient.
Combing the two statements => k must have only 2 and 5 as its prime factors
Hence 14k will have 3 prime factors
Hence Sufficient
Hence C.


40)Data Sufficiency->If x is an integer, is x^3 divisible by 9?

(1) x^2 is divisible by 9.
(2) x^4 is divisible by 9.
Show Spoiler
D.
Property in action -> \(X\) and \(X^n\) always have the exact same prime factors.


41)Data sufficiency->How many different prime numbers are factors of positive integer n?

(1) 4 different prime numbers are factors of 2n
(2) 4 different prime numbers are factors of n^2
Show Spoiler
B.
The question is just asking us the prime factors for n.
Statement 1-> From this statement we can conclude that n can have either 4 or 3 prime factors.
Hence Not sufficient
Statement 2-> Using the Property => X and X^n always have the exact same prime factors => As n^2 has 4 prime factors => n myst have 4 prime factors too.
Hence B.


42)If x and y are positive integers, are they consecutive?

1) x+y=3
2) x-y=1
Show Spoiler
D.


43)Data Sufficiency->Is the integer r divisible by 3?

(1) r is the product of 4 consecutive integers.
(2) r < 25
Show Spoiler
A.
Property in action -> Product of n consecutive integers is always divisible by n factorial.
So from statement 1-> r=24k for some integer k.


44)Data Sufficiency->Is x^2*y^4 an integer divisible by 9 ?

(1) x is an integer divisible by 3.
(2) xy is an integer divisible by 9.
Show Spoiler
E.
Using test cases can be extremely helpful here.
Clearly statement 1 and 2 are not sufficient on their own.
Combining them lets make test cases =>
Case 1=>
x=3
y=3
x^2*y^4 will be divisible by 9.
Case 2=>
x=18
y=1/2
x^2*y^4 will not be divisible by 9.
Hence E.


45)Data Sufficiency->For positive integers x and y, x^2 = 350y. Is y divisible by 28?
(1) x is divisible by 4.
(2) x^2 is divisible by 28.

Show Spoiler
A.
We are told that x and y are both positive integers.
x^2=350y
As x is an integer => x^2 must be a perfect square.
x^2=350y=2*5^2*7*y
Hence the general expression for y would be -> 2*7*p^z
where p is an integer and z is even.
We are asked if y is divisible by 28 or not.
Clearly y is already divisible by 7 and 2
Hence for y to be divisible by 28=> p must be even too.
Statement 1->
x is divisible by 4
Hence x^2 must have 2^4.
As x=2*5^2*7*y => For x^2 to have a 2^4 => y must have a 2^3.
Hence y must have a 2^3 and it already has a 7.
Thus y must be divisible by 28
Hence sufficient.
Statement2->
Here lets use test cases
y=2*7 =>y will not be divisible by 28
y=2*7*2^2 => y will be divisible by 28.
Hence not sufficient.
Hence A.


46)Data Sufficiency-> How many Prime factors does positive integer \(n\) have ?
A)\(n^3\) has 7 prime factors.
B)\(√n\) has 7 prime factors.
Show Spoiler
D.
\(X\) and \(X^n\) always have exact same prime factors.
In other words => "Exponentiation does not change the prime factors of any positive integer X."


47)If x and y represent the number of factors of 90 and 147 respectively.What is the value of x-y?
Show Spoiler
6.
\(90=2*3^2*5\)
Number of factors of \(90 =>2*3*2=> 12\)
So x=12
\(147=3*7^2\)
Number of factors of \(147=>2*3=>6\)
So y=6
Hence x-y=12-6=6.


48)If n is the product of all the integers from 1 to 10 exclusive,how many factors does n have?
Show Spoiler
160.

49)How many even factors does 21600 have?
A)32
B)42
C)60
D)25
E)52
Show Spoiler
C.
21600=2^5*3^3*5^2
For even factors -> The power of 2 should at-least be 1.
Hence possible cases -> 5*4*3=>60.
Additionally, for odd factors -> The power of 2 must be zero.
Possible cases -> 1*4*3=> 12.

50)How many positive odd divisors does 540 have?
A)6
B)8
C)12
D)15
E)24
Show Spoiler
B.
Here 540=2^2*3^3*5
Odd factors => 1*4*2=>8
Additionally Even factors =>2*4*2=>16
Total Factors =>3*4*2=>24.

51)How many of the factors of 72 are divisible by 2?
A)4
B)5
C)6
D)8
E)9
Show Spoiler
E.
72=2^3*3^2
For Even divisors ->Power of 2 must be at-least 1.
Possible cases=>3*3=> 9.
Hence E.

52)The number 100^2 has how many factors?
A) 28
B) 25
C) 24
D) 20
E) 16
Show Spoiler
B.

53)How many factors of 330 are odd numbers greater than 1?
A)8
B)7
C)6
D)5
E)4
Show Spoiler
B.
330=2*5*3*11
For Odd factors => Power of 2 must be zero
Possible cases -> 1*2*2*2=>8
So 330 has 8 odd divisors including one.
Excluding one => 330 must have 7 odd divisors.
Hence B.

54)If \(N=2^7*3^5*5^6*7^8\). How many factors of \(N\) are divisible by 50 but NOT by 100?
A)240
B)345
C)270
D)120
E)None of these
Show Spoiler
C.
For factors to be divisible by 50 and not divisible by 100 -->
Allowed powers of 2->one
Allowed powers of 5->5
Allowed powers of 3->6
Allowed powers of 7->9
Hence possible cases ->1*5*6*9=>270.
Hence C.


55)Data Sufficiency->How many positive factors does the positive integer x have?

(1) \(x\) is the product of 3 distinct prime numbers.

(2) \(x\) and \(3^7\) have the same number of positive factors.
Show Spoiler
D.

56)Data Sufficiency->How many factors does \(x\) have, if \(x\) is a positive integer?

(1) \(x = p^n\), where \(p\) is a prime number

(2) \(n^n = n + n\), where \(n\) is a positive integer

Show Spoiler
C.
From statement 1-> Number of factors of x will be (n+1)
Since we don't have any clue of n => not sufficient.
From statement 2-> n must be 2 as no other value of n will satisfy the equation.
Since we don't have any clue of x => not sufficient.
Combining the two statements => Number of factors of x-> n+1=2+1=3
Hence C.

57)Data Sufficiency->If 2,3,5 are the only prime factors of a positive integer p,what is the value of p?
A)p>100
B)p has exactly 12 factors including 1 and 12.
Show Spoiler
C.
Since 2,3,5 are the only prime factors of p=> The general expression for p will be \(2^a*3^b*5^c\) where a,b,c are positive integers.
Statement 1->p>100.
There exist infinite such cases.
In-fact there are only a few cases where p will be less than 100.
Hence not sufficient.
Statement 2->
Factors =12
Hence (a+1)*(b+1)*(c+1)=12=2*2*3
NOTE-> We can't arrange 12 as 1*1*12 or 2*6*1 or 4*3*1 as a,b,c are positive integers.
Hence a,b,c => (1,1,2) or (1,2,1) or (2,1,1)
Posible values of p=> 150,90,60
Hence not sufficient.
Combining the two statements => p must be 150.
Hence C.


This is probably the hardest Question in this set.

58)Data Sufficiency->Is the positive integer P prime?
(1) 61<P<67
(2) P is not divisible by 2.
Show Spoiler
A.
There are no prime numbers between 61 and 67.

59)Which of the following statements is/are true?
A)A perfect square always has an odd number of factors.
B)If a given integer has an odd number of factors then that integer must be a perfect square.
Show Spoiler
Both A and B are true.
Two Important Properties ->
A perfect Square always has an odd number of factors
If a number has an odd number of factors => It must be a perfect square.

60)How many factors does 64 have?
A)8
B)6
C)7
D)6
E)4
Show Spoiler
C.
Method 1=> 64=2^6
Hence it will have 7 factors
Hence C.
Method 2=> 64 is a perfect square =>Thus, It will always have an of Odd number of factors.
The only answer choice that is odd is C.

61)Data Sufficiency->Is the positive integer \(n\) divisible by \(6\)?
(1) \(\frac{n^2}{180}\) is an integer.

(2) \(\frac{144}{n^2}\) is an integer

Show Spoiler
A.
For n to be divisible by 6 => it must be divisible by both 2 and 3.
Statement 1->
As n^2 is divisible by 180
n^2 must have both 2 and 3 as its prime factors.
But X and X^p always have the exact same prime factors.
Hence n must also have both 2 and 3 as its primes too.
Hence n must be divisible by 6=> Sufficient
Statement 2->
Using test cases ->
n=1=> \(\frac{144}{n^2}\) will be an integer=>NO n is not divisible by 6.
n=6=>\(\frac{144}{n^2}\) will be an integer => YES n is divisible by 6.
Hence not Sufficient.
Hence A.

62)If a positive integer x has 5 prime factors, how many prime factors does 3x have?
A) 5
B) 6
C) 8
D) 15
E) cannot be determined
Show Spoiler
E.
Let \(x=(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Where \(p1,p2,p3,p4,p5\) are prime factors of x and a,b,c,d,e are positive integers.
\(3x=3*(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Hence if any one of p1,p2,p3,p4,p5 is 3 => 3x will have 5 prime factors.
Else => 3x will have 6 prime factors.
Hence 3x may have 5 or 6 prime factors.
So there is no unique answer.
Hence E.

63)How many positive distinct prime factors does 5^20 + 5^17 have?
A) One
B) Two
C) Three
D) Four
E) Five
Show Spoiler
D.

\(5^{17}+5^{20}=> 5^{17}(5^3+1) = 5^{17}*126=> 2*3^2*5^{17}*7\)
So it has 4 prime factors.
Hence D

64)How many prime numbers exist between 200 and 220?
(A) None
(B) One
(C) Two
(D) Three
(E) Four
Show Spoiler
B.
Property->In order check whether a given number is prime or not => We just need to divide it by all the prime numbers less than or equal to the square root of that number.
Highest value in this set-> 220
\(√220\)=14.something
Hence we need to check the divisibility with primes less than 14.something
=> {2,3,5,7,11,13}
All primes >2 are odd
So to shorten our calculation we just need to check the odd numbers.
201-> Divisible by 3
203->Divisible by 7
205->Divisible by 5
207->Divisible by 3
209->Divisible by 11
211-> PRIME=> Not divisible by any prime numbers in our list.
213-> Divisible by 3
215->Divisible by 5
217->Divisible by 7
219->Divisible by 3
Hence only 211 is a prime number.
Hence B.

65)Data Sufficiency->If y is a positive odd integer less than 33,does y have a factor p such that 1<p<y?
A)y+2 is divisible by 3.
B)Units digit of y is 7.
Show Spoiler
C.
Note-> Does y have a factor p such that 1<p<y can be rephrased as -> Is y prime?
The Question is just asking us if y prime or not.
Statement 1-> y=> {1,4,7,10,13,16,19,22,25,28,31}
Hence not sufficient.
Statement 2->
y can be ->
7-> Prime
17-> Prime
27-> Composite
Hence not sufficient
Combing the two statements -> y=7 is the only possible value that satisfies both the statements.
Hence y=7
So y is prime
Hence C.

66)If x is a positive integer less than 100, for how many values of x is x/6 a prime number?
A)2
B)6
C)8
D)13
E)17
Show Spoiler
B.

67)Data Sufficiency->If n is an integer between 10 and 99, is n < 80?
(1) The sum of the two digits of n is a prime number.
(2) Each of the two digits of n is a prime number.
Show Spoiler
B.
We can use test cases to arrive at an answer here.
Statement 1->
n=92=> sum=11 which is prime number.
So n is not less than 80.
n=12=>sum=3 which is a prime number.
So n is less than 80.
Hence not sufficient.
Statement 2->
Both the digits of n are prime.
Allowed tens digit of n->{3,5,7}
So n must be of the form 3X or 5X or 7X where X is the units digit of n.
So,n can never be greater than 80 as both 8 and 9 are non prime numbers.
Hence sufficient.
Hence B.
Source->Gmat-Prep

68)If n is a positive integer, which of the following statements are correct?
A)All the prime numbers greater than 3 can be represented in the form 4n +1 or 4n + 3.
B)All numbers of the form 4n + 1 and 4n + 3 are prime numbers.
C)All the prime numbers greater than 3 can be represented in the form 6n – 1 or 6n + 1.
D)All numbers of the form 6n + 1 and 6n - 1 are prime numbers.
Show Spoiler
A and C are correct.
An important property of prime numbers ->Every prime number greater than 3 can be written as=>
6n+1 OR 6n-1
6n+5 OR 6n-5
4n+1 OR 4n-1
4n+3 OR 4n-3
This can be verified taking the first few primes.
But alternatively every number of these forms isn't prime.
E.g-> 25=6n+1 or 4n+1 and it isn't prime.

69)Data Sufficiency->If p is an integer, is p + 4 a prime number less than 50?
(1) p + 1 is a prime number.
(2) p is a prime number.
Show Spoiler
C.
Note=>2 and 3 are the only consecutive integers that are also prime numbers.

70)How many odd factors does the number 2100 have?
A)36
B)24
C)12
D)8
E)cannot be determined.
Show Spoiler
C.
\(2100=2^2*3*5^2*7\)
For odd factors => Power of 2 must be zero.
Possible cases -> \(1*2*3*2=>12\).
Hence C.

71)Data Sufficiency->If p is an integer greater than 1, is p a prime number?
(1) p is a factor of 13.
(2) p is a factor of 78.
Show Spoiler
A.
We are told that p is an integer greater than 1 and we are asked if p is a prime number or not.
Statement 1-> p is a factor of 13.
But 13 has got only 2 factors as it is a prime number.
Factors => 1 and 13
so p must be either 1 or 13.
But p>1
Hence p must be 13.
So p is definitely a prime number.
Sufficient.
Statement 2->
p is factor of 78.
Lets use test cases=>
p=2->2 is a factor of 78 and 2 is a prime number.
p=78->78 is a factor of 78 and clearly 78 isn't a prime number .
Hence not sufficient.
Hence A.
Source->GMAT-prep

72)If the number 13 completely divides \(x\), and \(x = a^2 * b\), where a and b are distinct prime numbers, which of these numbers must be divisible by 169?
A)\(a^2\)
B)\(b^2\)
C)\(a*b\)
D)\(a^2*b^2\)
E)\(a^3*b\)
Show Spoiler
D.
As 13 is a prime factor of x => either a or b will be 13.
Hence a^b*b^2 will always be divisible by 169

73)Data sufficiency->If p is an integer and p>1,is p a prime number?
A)p has p factors.
B)p is a factor of 26.
Show Spoiler
A.
There are only two numbers for which the number of factors = number itself.
They are ->{1,2}
As p>1 => p must be 2.
So p=2
Clearly p is a prime number.
Hence Sufficient.
Statement 2->Using test cases
p=2-> 2 is a factor of 26 and 2 is a prime number.
p=26-> 26 is a factor of 26 and 26 is not a prime number.
Clearly not sufficient.
Hence A.


74)Data Sufficiency->Is the number of distinct prime factors of the positive integer N more than 4?
(1) N is a multiple of 42.
(2) N is a multiple of 98
Show Spoiler
E.

75)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/30 is an integer
(2) k < 100
Show Spoiler
C.

76)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/60 is an integer
(2) k < 100
Show Spoiler
C.

77)Data Sufficiency->If A and B are positive integers, does A have more prime factors than B?
(1) B is the square root of A
(2) A ÷ B is an integer
Show Spoiler
A.
Property in action->\(X\) and \(X^n\) always have the exact same prime factors.

78)What is the greatest prime factor of \(12^3 – 96\) ?
A)2
B)3
C)7
D)17
E)19
Show Spoiler
D.
\(12^3-96=2^6*3^3-2^5*3=2^5*3(2*3^2-1)=2^5*3(18-1)=2^5*3*17\)
Clearly here the greatest prime Factor is 17.

79)What is the greatest prime factor of 4^17 - 2^28?
A)2
B)3
C)5
D)7
E)11
Show Spoiler
D.

80)What is the total number of distinct prime factors of 28980?
A)2
B)3
C)4
D)5
E)6
Show Spoiler
D.
\(28980=2^2*3^2*5*7*23\)
Hence it has 5 prime factors.
Hence D.

81)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
Show Spoiler
D.
using the identity a^2-b^2 = (a+b)(a-b)=> (3^3)^2 -1^2 = 28*26 = 2^3*7*13
Clearly the greatest prime factor will be 13.

82)Find the number of factors of 180 that are in the form (4*k + 2), where k is a non-negative integer?
A)1
B)2
C)3
D)4
E)6
Show Spoiler
E.
Method 1->
180 is a relatively smaller number=>
Factors can be seen as follows ->
1*180
2*90
3*60
4*45
5*36
6*30
9*20
10*18
12*15
15*12(The repetition has begun.
Hence checking the factors we get => There are 6 factors of the form 4k+2 for k≥0
Method 2->
180=2^2^3^2*5
Factors of the form -> 4k+2 => 2(2k+1)=> 2*odd number.
2*1
2*3
2*3^2
2*5
2*3*5
2*3^2*5
Hence six values.
E.


83)Data Sufficiency->If A and B are positive integers, does B have more prime factors than A?
(1) B is the square root of A
(2) A ÷ B is an integer
Show Spoiler
D.
X and X^n have the exact same primes.
So Statement 1 is sufficient.
As for statement 2-> If A/B is an integer => Every prime factor of B must be a prime factor of A.
Hence B can never have more prime factors than A.
Hence Sufficient.
Hence D.

84)If N = 10! then what is the minimum number which should be multiplied with N to make it a perfect square?
A)7
B)14
C)21
D)28
E)36
Show Spoiler
A.
10!=1*2*3*4*5*6*7*8*9*10=> 2^8*3^4*5^2*7
In a perfect square the Power of every prime must be even.
Thus =>Smallest number to be multiplied will be 7.
Hence A.

85)If N = \(2^2*3^3*5^5\), find the total number of odd and even factors of N.
A)24,96
B)15,30
C)24,48
D)15,96
E)15,48
Show Spoiler
C.
For even factors => Power of 2 must be at-least 1.
Hence possible cases => 2*4*6 =>48
odd factors => Power of 2 must be zero.
Possible cases =>1*4*6=>24

Hence 48,24 is the correct option.

86)Which of the following numbers is a prime number?
A)2343
B)3457
C)4689
D)7731
E)9861
Show Spoiler
B.
Every number except the second one is dividable by 3 so they can never be prime.
Hence B must be a prime number.
Alternatively prime numbers can be written as either 6n+1 OR 6n-1
Only B can be written as 6n+1
One of the other options can be written as 6n+1 or 6n-1.
Hence B is a prime number.

87)The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?
(A) \(10^9\)
(B) \(10^8\)
(C) \(10^7\)
(D) \(10^6\)
(E) \(10^5\)
Show Spoiler
Product=>2*3*5*7*11*13*17*19=> 10*20*150*300=>9000000=> Close to 1 crore or 10^7 then it is close to 10 lakhs or 10^6
Hence C.
Source->Official Guide.

88)If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?
(A) four
(B) five
(C) six
(D) seven
(E) eight
Show Spoiler
A.
Source->Official Guide.

89)Find the number of factors of 240 that are in the form (4*p + 2), where p is a non-negative integer?
A)1
B)2
C)3
D)4
E)5
Show Spoiler
D.

90)Data Sufficiency->What is the value of the integer p ?
(1) Each of the integers 2, 3, and 5 is a factor of p.
(2) Each of the integers 2, 5, and 7 is a factor of p.
Show Spoiler
E.
Basically n can be any integer that has 2,3,5,7 as its prime factors.
Eg-> 2*3*5*7
OR
2*3*5*7*13
Etc.
Hence E.
Source->Official Guide.

91)Data Sufficiency->Does the integer k have at least three different positive prime factors?
(1) k/15 is an integer.
(2) k/10 is an integer.
Show Spoiler
C.
Statement 1->3 and 5 must be the prime factors of K.
Statement 2-> 2 and 5 must be the prime factors of K.
combing them -> 2,3 and 5 must be the prime factors of K.
Hence C.
Source->Official Guide.

92)If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18
Show Spoiler
C.
Source->Official Guide.

93)If n = 20! + 17, then n is divisible by which of the following?

I. 15
II. 17
III. 19

(A) None
(B) I only
(C) II only
(D) I and II
(E) II and II
Show Spoiler
C.
Source->Official Guide.

94)If 3 < x < 100, for how many values of x is x/3 the square of a prime number?
(A) Two
(B) Three
(C) Four
(D) Five
(E) Nine
Show Spoiler
B.
Source->Official Guide.

95)How many integer values are there for x such that 1 < 3x + 5 < 17?
A)Two
B)Three
C)Four
D)Five
E)Six
Show Spoiler
D.
x->(-1.3,4)
So possible integer values for x =>-1,0,1,2,3
Thus,five values for x are possible.
Hence D.

96)For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301?
A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150
Show Spoiler
B.
Source->Oficial Guide


97)If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S
Show Spoiler
C.
The easiest way here to pick options.
Let n=1
Sum=1
Sum of evens = 2
Hence for S=1=> Sum of evens = 2
Only Option C fits that case.
NOTE-> Sometimes while using test cases more than one value may satisfy.In that case we must use some additional test cases to arrive at the correct answer.
Source->2011 Official Guide.


98)How many prime numbers between 1 and 100 are factors of 7,150 ?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
Show Spoiler
D.
The question is asking us in a fancy way to get the prime factors of 7150 between 1 and 100.
Breaking down 7150=> 2*5^2*11*13
Hence It has 4 prime factors.
Hence D.
Source-> Official Guide.

99)If n is a prime number between 0 and 100, how many positive divisors does n^3 have?
A)1
B)2
C)3
D)4
E)5
Show Spoiler
D.

100)If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A)2
B)5
C)6
D)7
E)14
Show Spoiler
E.
Note-> For any perfect square -> Power/exponent of each prime factor must be even.
Source-> Official Guide




Note-->PM me if you find any answers/solutions inaccurate/inadequate.

Any Feedback would be Appreciated.
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Originally posted by stonecold on 23 Apr 2016, 06:17.
Last edited by stonecold on 24 Jan 2017, 20:46, edited 11 times in total.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  26 Nov 2016, 07:58
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Day-3


Mock Test 3



Number of Questions->85

Topic Covered-> Evens/odds

Source-> This Quiz contains questions from varied sources along with quite a few self-made questions.

1)Which of the following statements must be true?
A)Any integer can be either even or odd but not both.
B)2n is always even.
C)2n+1 is always odd.
D)Zero is an even integer.
E)The sum of even number of odd numbers is always is always even.
F)The sum of odd number of odd numbers is always odd.
G)If any factor of an integer is even,then the integer itself would be even.
Show Spoiler
A,D,E,F,G
Note -> Since n can be an integer or a non integer -> 2n may be even or odd.
Similarly 2n+1 may be even or odd.



2)Data Sufficiency->If n is a positive integer,is n divisible by 2?
A)7n-8 is divisible by 20.
B)3n^2+2n+5 is a prime number.
Show Spoiler
D.



3)If x,y,z are integers and p=x*y*z,if p odd?
A)x is odd.
B)z is even.
Show Spoiler
B.



4)If x and n are non negative integers,then which of the following statements must be true?
A)If x is even then x^n will always be even.
B)If x is odd then x^n will always be odd.
Show Spoiler
B.
Notice that n can be zero too.
Although x and x^n will always have the same even/odd nature,we must make sure than the exponent is non zero.
Anything^zero=1 which is always odd.



5)Data Sufficiency->If \(A\) and \(B\) are integers,is \(A*B^4\)even?
A)\((2A+B)^3\) is even.
B)\(A+3B\) is odd.
Show Spoiler
D.



6)If w,x,y,z are consecutive positive integers then which of the following statements will always be odd?
I) wx+(y+2)^2*z
II) w^x+z^y
III) 2w^3-3x^2-5y+6z^2
A)I
B)II
C)III
D)I and II
E)II and III
Show Spoiler
E.



7)Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88
Show Spoiler
D.
Essentially this is testing our knowledge on Even/odd numbers.
If sum of 2 primes is even => 2 must not be either of them.
If sum of 2 primes is odd => 2 must be one of them.

Hence as 79 =2+77 =>79 can never be written as sum of 2 prime numbers.

Hence D.



8)Data Sufficiency->If \(x\) and \(y\) are integers, is \(3*x^4 + 4*y\) even?
(1) \(x^3\) is even.
(2) \(y^{2x} + 3\) is even.
Show Spoiler
A.



9)If x and y are integers is y odd?
A)y^2x +8 is odd.
B)y^7 is odd.
Show Spoiler
B.
Note-> Statement 1 is not sufficient as we gotta consider the case of x=0.



10)Data Sufficiency->If \(X\) is a positive integer, is \(X^2 + 1\) an odd number?
(1) \(X\) is the smallest integer that is divisible by all integers from 11 to 15, inclusive.
(2) \(3^X\) is an odd number.
Show Spoiler
A.



11)Data Sufficiency-> If a, b, and k are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( k^3 + 3k^2 + 3k + 6)
(2) b = (k^2 + 4a +5)
Show Spoiler
C.



12)Data Sufficiency->If x, y, and z are positive integers, where x is an odd number and z = x^2 + y^2 + 4. Is y^2 divisible by 4?
(1) Z = 8k -3 where k is a positive integer.
(2) When (z-x+1) is divided by 2, it leaves a remainder.
Show Spoiler
D.



13)If a, b, c, d, and e are integers and the expression \(\frac{a*b^2*c^2}{d^2*e}\) gives a positive even integer, which of the following options must be true?
I. \(a*b*c\) is even
II. \(\frac{a}{e}\) is positive
III. \(\frac{a}{d^2}\) is positive.

A)I
B)II
C)I and II
D)I and III
E)I,II,III
Show Spoiler
C.



14)Data Sufficiency->If y is a positive integer is y divisible by 21?
A)y^2 is divisible by 21.
B)y^3 is divisible by 63.
Show Spoiler
D.



15)If r and s are positive integers, and \(r^2\) + \(\frac{r}{s}\) is an odd integer, which of the following cannot be even?
A)\(3*r+2*s\)
B)\((r-1)*(s+2)\)
C)\(r^{s+1}+s^r\)
D)\(r^3+3\)
E)\(s^4+4\)
Show Spoiler
D.



16)Data Sufficiency->If x and y are integers, is y even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
Show Spoiler
E.



17)Data Sufficiency->If x and y are integers, is x even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
Show Spoiler
C.



18)If A is a positive integer, then which of the following statements is true?
1. A^2 + A -1 is always even.
2. (A^4+1)(A^4+2) + 3A is even only when A is even.
3. (A-1)(A+2)(A+4) is never odd.

A)1
B)2
C)3
D)1 and 2
E)2 and 3
Show Spoiler
E.



19)Data Sufficiency-->If the positive integer N divisible by 2?
A)11n+8 is divisible by 4.
B)5n^2+2n+13 is a prime number.
Show Spoiler
D.



20)Set S is given as S = {1,3,5,7,9,11,13,15,17}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 18?
A)zero
B)two
C)three
D)six
E)nine
Show Spoiler
A.
Notice that all the integers in the set are odd
So whenever we pick any 3 numbers => Sum will be odd+odd+odd=> odd
Hence sum will never be 18 which is even.
Hence A.



21)Data Sufficiency ->If A and B are positive integers,is the Product A*B even?
A)A is odd
B)B^43 is even
Show Spoiler
B.



22)Which of the following statements must be true?
The product of first 100 prime numbers is even.
The sum of first 100 prime numbers is odd.
The sum of first five non-negative even numbers is divisible by both 4 and 5.

A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
Show Spoiler
E.
Properties in action => Sum of odd number of odd numbers -> Odd
Sum of even number of odd numbers is even.
Two is the only even prime.



23)If p,q,r,s are consecutive positive integers,then which of the following statements must be odd ?
I)p*s + [(q+14)^2] *r
II)p^r + s^q
III)26*p^3 + 3*q^23 + 5*r^21 + 28*s^2

A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III
Show Spoiler
D.



24)Data Sufficiency->If x, y, z and w are positive integers, is x odd?
(1) 7x + 8y + 4z + 5w is odd.
(2) 3x + 2y + 8z + 2w is even.
Show Spoiler
B.


25)7)Data Sufficiency->Is z even?
(1) \(\frac{z}{2}\) is even.
(2) 3z is even.
Show Spoiler
A.
Notice in statement 2 z may or may not be an integer.



26)Data Sufficiency->Is z even?
(1) 5z is even.
(2) 3z is even.
Show Spoiler
C.



27)Data Sufficiency->If \(x\) and \(y\) are positive integers and \(x\) is odd, is \(x*y\) even?
(1) \(x^3*y = 6*a^3 + 23\) where a is a positive integer.
(2) \(x^2+y = 3*k + 7\) where \(k\) is a positive integer.
Show Spoiler
A.



28)Data Sufficiency -> Is z even?
(1) 5z is even.
(2) 15z is even.
Show Spoiler
E.



29)Data Sufficiency->If a and b are integers, is b even?
(1) 3a + 4b is even.
(2) 3a + 5b is even.
Show Spoiler
C.



30)If n is an integer, then which of the following statements is/are FALSE?
I)\(n^3 – n\) is always even.
II)\(8n^3 +12n^2 +6n +1\) is always even.
III)\(√ (4n^2 – 4*n +1)\) is always odd.

A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III

Show Spoiler
B.
Notice that for statement 3-> 4n^2-4n+1 is (2n-1)^2



31)Data Sufficiency ->If t is a positive integer, is t^3 + 1 an odd number?
(1) t is the smallest integer that is divisible by all integers from 21 to 25, inclusive.
(2) 5^t is an odd number.
Show Spoiler
A.



32)Data Sufficiency ->If x and y are integers, is 31x^4 + 4y even?
(1) \(x^{13}\) is even.
(2) \(y^{12x}\) + 3 is even.
Show Spoiler
A.
Also notice that in statement 2,its not a necessity that y will be odd.If x=0 then y can be even or odd.This fact is useless in this questions as we don't want the even/odd nature of y,rather we are only interested in the even/odd nature of x.



33)Data Sufficiency ->If p, q, and r are positive integers, where p is an odd number and r = p^2 + q^3 + 4. Is q^3 divisible by 8?
(1) r = 18k -5 where k is a positive integer
(2) When (r-p+13) is divided by 2, it leaves a remainder.

Show Spoiler
D.
Notice that this question is just asking us for the even/odd nature of q.
If q is even => q^3 must be divisible by 8.



34)Data Sufficiency ->If a, b, and t are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( t^13 + t^12 + 3t+ 6)
(2) b = (t^2 + 4a +5)

Show Spoiler
C.
This is a typical question in which the even/odd nature of two integers is connected via a third integer.



35)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is odd.
Show Spoiler
C.
NOTE-> B is alone not sufficient as x can be zero too.
And anything ^0 is one which is always odd.



36)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is even.
Show Spoiler
B.
Notice that if y^2x is even => x can never be zero => y must always be even.



37)Data Sufficiency->Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where n is a positive number.
Show Spoiler
E.
Notice we are not told if p is an integer or not.
E.g=> p=√2 for n=√2/2 and clearly p^2 is not divisible by 4.



38)Data Sufficiency->If p is an integer,Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where is n is a positive integer.
Show Spoiler
D.



39)If a and b are positive integers, and a^3 + a^2/b is an odd integer, which of the following cannot be even?
A)31a + 12b
B(a-1)(s+10)
C)ab+21b
D)a^3 +7
E)b^4 +4
Show Spoiler
D.
Notice that if a^3 + a^2/b is odd -> a must be even and b must be even too.



40)Set S is given as S = {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 68?
A)zero
B)1
C)19
D)38
E)19!

Show Spoiler
A.
Notice that they are all odd.
Hence, sum of any three integers picked from this set will be odd + odd +odd -> odd
Hence sum can never be 68.
Hence A.



41)Data Sufficiency->If a+b is even, is b an integer?
1) a-b is even.
2) a+3b is even.

Show Spoiler
D.
Hint => If 2x is even => x must be an integer.



42)Which of the following statements must be true?
I)The product of first 249 prime numbers is even.
II)The sum of first 249 prime numbers is even.
III)Sum of any 13 prime numbers greater than 100 is always odd.

A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
Show Spoiler
E.
Two properties are in action here.
ONE-> Sum of even number of odd numbers is even and the Sum of odd number of odd numbers is odd.
TWO-> All primes numbers greater than 2 are odd.



43)State True/False->The sum of first 5 non negative of 5 is divisible by both 25 and 50.
A)True
B)False
Show Spoiler
True.
Sum=0+5+10+15+20=50.
50 is divisible by both 25 and 50.


44)Data Sufficiency-> If z is an integer, is z even?
(1) z/2 is not an odd integer.
(2) z + 5 is an odd integer.
Show Spoiler
B.
We need the even odd nature of integer z

Statement 1->
z=5
z=4

Taking these two test cases we can say that this is an insufficient statement

Statement 2->
z-odd=odd => z=> odd+odd=even
Hence sufficient

Hence B.

45)Data Sufficiency->Is the integer a even?
(1) a is divisible by 7.
(2) a is divisible by 9.
Show Spoiler
E.



46)Data Sufficiency-> Is a even?
(1) 2a is even.
(2) √a is even.
Show Spoiler
B.



47)Data Sufficiency->Is n an even?
1) 3n is even.
2) 7n is even.
Show Spoiler
C.
Note that it is not given in the original question that n is an integer.



48)Data Sufficiency-> If n is an integer, is n even?
(1) 2n is divisible by 4.
(2) n^2 is even.
Show Spoiler
D.
Source-> GMAT-prep



49)If n is an integer, which of the following must be even?
(A) n+1
(B) n+2
(C) 2n
(D) 2n +1
(E) n^2
Show Spoiler
C.
Source->Official Guide.



50)Data Sufficiency->\(w\), \(x\), \(y\) and \(z\) are all integers. Is \(w*x*y*z\) even?
(1) \(w*x*y\) is odd.
(2) \(x*y – z\) is even.
Show Spoiler
C.




51)Data Sufficiency->If x, y, and z are all positive integers, is x + y + z even?
1) \(\frac{2x}{(y+z)}\) = odd.
2) \(\frac{x^2}{4yz}\) is an integer.
Show Spoiler
C.



52)Data Sufficiency->If x is an integer, is x even?
(1) ax is even, where a is an integer.
(2) x^2 - 1 is an odd integer.
Show Spoiler
B.



53)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an even integer.
Show Spoiler
E.



54)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an odd integer.
Show Spoiler
C.


55)Data Sufficiency->If x is a positive integer,is x even?
(1)x^2+y^2=98
(2)x is odd.
Show Spoiler
C.



56)Data Sufficiency->If x any y are positive integers,is x even?
(1)x^2+y^2=98
(2)x=y.

Show Spoiler
A.
The only value of a and b that will satisfy statement 1 is 7.
Hence a=b=7



57)Data Sufficiency->If x, y and z are integers and xy + z is an even integer, is x an even integer?
(1) xy + xz is an even integer.
(2) y + xz is an odd integer.
Show Spoiler
B.


58)If both x and y are positive even integers, then which of the following expressions must also
be even?

I. \(y^{x − 1}\)

II. \(y – 1\)

III. \(\frac{x}{2}\)

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Show Spoiler
A.



59)If both x and y are positive integers and y is even,then which of the following expressions must also
be even?

I. \(y^{x − 1}\)

II. \(x^{y-1}\)

III. \(\frac{[fraction]y^2}{2}[/fraction]\)

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
Show Spoiler
C.



60)If a is an odd integer and b is an even integer, which of the following CANNOT be true?
A. a + b is an odd integer.
B. a is a factor of b.
C. b is a factor of a.
D. a^b is an odd integer.
E. b^a is an even integer.
Show Spoiler
C.
An even number can never divide an odd number.
Hence C.



61)If x is a positive odd integer and y is a negative even integer, which of the following must be true?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
Show Spoiler
C.



62)If x is a positive odd integer and y is a negative even integer, which of the following must be false?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
Show Spoiler
B.



63)If x is an even integer and y is an odd integer, which of the following must be an even integer?
A. y/x
B. x+y
C. 3x + 2y
D. 3(x + y)
E. 2y/x
Show Spoiler
C.



64)If positive integers x and y are not both odd, which of the following must be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
Show Spoiler
A.
Source->Official Guide



65)If positive integers x and y are not both even, which of the following must be even?
A)xy
B)x+y
C)x-y^2
D)xy(x+y)
E)x/y
Show Spoiler
D.



66)If positive integers x and y are not both odd, which of the following can never be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
Show Spoiler
E.



67)If x and y are integers and x + y = 5, which of the following must be true?
A) x and y are consecutive integers.
B) If x < 0, then y > 0.
C) If x > 0, then y < 0.
D) Both x and y are even.
E) Both x and y are less than 5.
Show Spoiler
B.
We need to check as to which of the statement is always true.
So if we can find a case that makes a statement false we can react the statement.

Statement 1->
-1,6 => Rejected.
Statment 2->
Always true.
Statement 3->
2,3=> Rejected
Statement 4->
2,3 => Rejected.
Statement 5->-1,6 => Rejected.
Hence B.



68)If a and b are odd integers, which of the following must be an even integer?
A. a(b – 2)
B. ab + 4
C. (a + 2)(b – 4)
D. 3a + 5b
E. a(a + 6)
Show Spoiler
D.



69)If p is an even integer and q is an odd integer, which of the following must be an odd integer?
A. p/q
B. pq
C. 2p+q
D. 2(p+q)
E. 3p/q
Show Spoiler
C.



70)If a and b are even integers, which of the following is an odd integer?
A. ab + 2
B. a(b − 1)
C. a(a + 5)
D. 3a + 4b
E. (a + 3)(b − 1
Show Spoiler
E.



71)If a is an even integer and b is an odd integer which of the following must be an odd integer :
A. a/b
B. ab
C. 2a+b
D. 2(a+b)
E. 3a/b
Show Spoiler
C.



72)If x and y are integers and x^2*y is a negative odd integer, which of the following must be true?

I. xy^2 is odd.
II. xy is negative.
III. x + y is even.

A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III
Show Spoiler
D.
We are told that x^2*y<0
Hence x≠0 and y≠0
So as x^2 is always greater than 0.
Hence y<0.
Also as x^2*y is odd => x and y must be both odd.

Option 1->xy^2=> odd*odd^2=> odd.
True.
Option 2-> xy will be negative for x being positive and xy will be positive for x being negative.
Hence this statement is not always true.
Option 3->
x+y=> odd+odd=> even.
Hence this is True.

So the correct answer is D.



73)If x is an odd integer and y and z are even integers, which of the following CANNOT be an integer?
A. y/z
B. x/y
C. z/x
D. yx/z
E. zx/y
Show Spoiler
B.
An odd number can never be divisible by even number.
Hence B.



74)X and Y are integers, X is even and negative, Y is odd and positive. Which of the following could be false?

1. (X + Y) is an odd number.
2. Y^(X + Y) is an integer.
3. X^Y is a positive number.

A. 2 only.
B. 3 only.
C. 1 and 3 only.
D. 2 and 3 only.
E. 1, 2 and 3.
Show Spoiler
D.



75)If x is an odd negative integer and y is an even integer, which of the following statements must be true?

I. (3x - 2y) is odd

II. xy^2 is an even negative integer

III. (y^2 - x) is an odd negative integer

A. I only
B. II only
C. I and II
D. I and III
E. II and III
Show Spoiler
A.



76)If x, y, and z are positive integers and x^2 = y^2 + z^2, which of the following must be true?

I. x > z
II. x = y + z
III. y^2 + z^2 is a positive integer

A. I only
B. II only
C. III only
D. I and II only
E. I and III only
Show Spoiler
E.



77)If x and y are positive and \(x^2+y^2\)=100, then for which of the following is the value of x+y greatest?
A)x=10
B)x=9
C)x=8
D)x=7
E)x=6
Show Spoiler
D.
We can use estimation to check the values of x+y in each option.

Option 1-> x=10 => y=0 --> Not allowed as y>0
Option 2-> x=9=> y=4.something => x+y=> 13.something
Option 3-> x=8=> y=6 =>x+y=14
Option 4-> x=7 => y=7.something => x+y=14.something
Option 5=> x=6=>y=8=> x+y=14

Hence clearly Option 4 will give us the greatest value of x+y
Hence D.


78)If x is an even integer, which of the following is an odd integer?
(A) 3x + 2
(B) 7x
(C) 8x +5
(D) x^2
(E) x^3
Show Spoiler
C.
eq


79)If x, y, and z are consecutive even positive integers, which of the following could be equal to x + y + z ?
A)141
B)200
C)318
D)391
E)412
Show Spoiler
C.
Let the 3 consecutive even integers be represents by -->
2n
2n+2
2n+4

Sum=> 6n+6-> 6(n+1)
Hence the sum must be a multiple of 6.
Only option that fits this criteria is C.

Hence C.



80)If x, y, and z are consecutive odd positive integers, which of the following could be equal to x + y + z ?
A)9
B)221
C)400
D)900
E)1200
Show Spoiler
A.



81)If w,x,y are consecutive odd positive integers with w<x<y, then which of the following maybe the value of y-x-w?
A)-4
B)-2
C)-1
D)0
E)3
Show Spoiler
C.
Source->GMAT-prep


82)If a is an odd integer, which of the following must be an even integer?
A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.
Show Spoiler
B.
Firstly -->Positive exponents does not affect the even/odd nature of any number
Option A-> a^4−a+1=> odd-odd+odd=> even+odd=> odd=> Rejected.
Option B->(a^4−a)(a+1/a)=> (a^3-1)*(a^2+a)=> (odd-odd)*(odd+odd) => even*even => even=> Acceptable.
Option C. a^4−a^3+a^2+2a=> odd-odd+odd+even=>even+odd=>odd=> Rejected.
Option D. (a^3+a^2+a)^2=>(odd+odd+odd)^2=>(even+odd)^2=>odd^2=> odd=>Rejected.
Option E. None of the above.=> Rejected.

83)If a and b are odd integers, which of the following must be an even integer?
A. a^2(b – 2)
B. ab + 40
C. (a + 22)(b – 42)
D. 31a + 51b
E. a(a + 16)
Show Spoiler
D.



84)If a and b are odd integers which of the following must be an even integer?
A. a/b
B. ab
C. (ab)^2
D. ab + b
E. (a + b)/b
Show Spoiler
D.



85)If x is even integer, which of the following must be an odd integer?
A. \(\frac{3x}{2}\)
B. \(\frac{3x}{2} + 1\)
C. \(3x^2\)
D. \(\frac{3x^2}{2}\)
E. \(\frac{3x^2}{2} + 1\)
Show Spoiler
E.
Source->GMAT-Prep.

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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  22 Nov 2016, 01:31
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Day-2



MOCK TEST-2


Advanced Quiz on Divisibility and Prime numbers.

Number of Questions-> 100.

Sources-> The following questions are from various sources and a few are self made.


101)If 2940 multiplied by k is a square of an integer,what is the smallest possible value of k?
A)3
B)6
C)15
D)21
E)cannot be determined.
Show Spoiler
C.

102)If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?
A)10
B)11
C)12
D)13
E)14
Show Spoiler
B.
Source-> Official Guide

103)n is a positive integer, and k is the product of all integers from 1 to n inclusive. If k is a multiple of 1440, then the smallest possible value of n is
A)8
B)12
C)6
D)18
E)24
Show Spoiler
A.

104)If x^2 is divisible by 216, what is the smallest possible value for positive integer x?
Show Spoiler
36.

105)If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?
A.3
B.9
C.12
D.15
E.20
Show Spoiler
C.
NOTE->If you choose A=> Have a relook at the Question."k is an even integer" is specified in the Question-stem.

106)If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?
(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight
Show Spoiler
C.
Easiest way is to put p=3 ->n=12 ->2,4,6,12 => Four even factors.
Alternatively -> \(n=2^2*p\) where p is an odd prime => for even factors => power of 2 must be at-least 1.
Possible cases => 2*2=> 4
Hence C.

107)If N = \(2^2 * 3^3 * 5^5\),How many factors of N are divisible by 5 but not divisible by 3.
A)10
B)12
C)14
D)15
E)20
Show Spoiler
D.
For factors to be divisible by 5 but not divisible by 3=> Power of 3 must be zero and power of 5 must be greater than or equal to 1.
Possible cases => 3*1*5->15
Hence D.

108)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
Show Spoiler
E.

109)Data Sufficiency->How many distinct factors does positive integer k have?
(1) k has more distinct factors than the integer 9 but fewer distinct factors than the integer 81.
(2) k is the product of two distinct prime numbers.
Show Spoiler
D.
Statement 1-> 9 has 3 factors and 81 has 5 factors => k must have 4 factors => Sufficient.
Statement 2-> k=p*q for prime p and q => Number of factors => 2*2=>4=> Sufficient.
Hence D.

110) What is the greatest prime factor of (2^29) - (2^26)?
A)2
B)4
C)7
D)8
E)3
Show Spoiler
C.

111)Data Sufficiency->How many different prime factors does positive integer n have?
(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors
Show Spoiler
D.
Statement 1->
Notice the boundary condition for n is given.
Since n is an integer => n^2 must be a perfect square.
n^2=> 49,64 or 81
So n can be 7 or 2^3 or 3^2
In each case -> n will have just one prime factor.
Hence Sufficient.

Statement 2->
8n^12 has twelve factors.
2^3*n^2 has 12 factors => This is only possible if n is a prime number.
Thus it will have only one prime factor.
Hence D.
Source-> GMATClub-Tests

112)What is the greatest prime factor of 9919?
A) 7
B) 13
C) 17
D) 97
E) 109
Show Spoiler
E.
9919=100^2-9^2=> 109*91=>109*7*13.
Notice that 109 is a prime number.
Hence the greatest prime factor ->109
Hence E.

113)If 5x^2 has two different prime factors, at most how many different prime factors does x have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Show Spoiler
B.
Two properties are been tested here =>
One-> \(X\) and \(X^n\) always have the exact same prime factors.
Two->When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
So \(x\) and \(x^2\) will have the exact same primes.
Two cases are possible here =>
Case 1=>
\(x\) has one prime factor which is not equal to 5.
So 5\(x^2\) will have 2 primes.
Case 2->
\(x\) has two prime factors,one out of which is 5 so that 5x^2 has two prime factors.

Hence almost \(x\) can have 2 prime factors.
Hence B.
Source->GMAT-prep

114)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
Show Spoiler
E.
12!11! + 11!10!=> 11!10![12*11+1]=>11!10!*133=>11!10!*7*19
Clearly the greatest prime factor will be 19.
Hence E.

115)Data Sufficiency->How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of \(2n\).
(2) 4 different prime numbers are factors of \(n^2\).
Show Spoiler
B.
Properties in action->
\(X\) and \(X^n\) always have the exact same prime factors.
When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
Statement 1=>
From this statement we can conclude that n can have 4 prime factors or 3 prime factors.
Hence not sufficient.
Statement 2=>
From this statement it is evident that n must have exactly 4 prime factors.
Hence Sufficient.
Hence B.
Source-> GMAT-Prep

116)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
Show Spoiler
3^6-1=> 27^2-1=> (27+1)*(27-1)=> 2^3*7*13
Clearly 13 is the greatest prime factor.
Hence D.

117)Data Sufficiency->If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}?
(1) mn is prime.
(2) m + n is even.
Show Spoiler
C.
In this Question,we are told that m and n are different positive integers and we are asked as to how many numbers out of {m, n, m + n} are prime.
Statement 1=>
m*n is a prime number
This can happen only when one out of m or n must be 1 and the other must be a prime number.
Case 1 => When an even prime i.e 2 is involved -> {1,2,3} => two numbers in this set will be prime.
Case 2=> When an odd prime is involved -> {1,odd-prime,even number >2}=> Only one prime number.
Hence not sufficient.
Statement 2->m+n is even
e.g=> {3,5,8} => two prime
or {2,4,6}=> None are prime => Not sufficient.
Combining the two statements => m+n is even and one out of the is one => The other must be odd prime.
Thus in the set => {1,Odd-prime,Even number>1}=> Only one number will be prime.
Hence C.

118)What is the greatest prime factor of 1+2+3+。。。+40?
A. 17
B. 29
C. 31
D. 37
E. 41
Show Spoiler
E.
Sum => 1+2+3....40=> 20*41 => 2^2*5*41
41 is clearly the greatest prime factor.
Hence E.

119)Data Sufficiency->How many positive factors does positive integer \(N\) have?
1) \(N^2\) has three positive factors.
2) \(2N\) has four positive factors.
Show Spoiler
A.
We are asked about the number of factors of a positive integer N.

Statement 1->
\(N^2\) has 3 factors
The only way \(N^2\) can have 3 factors is if N is a prime number.
So \(N\) must have 2 factors.
Hence Sufficient.
Statement 2->
\(2N\) has 4 Factors
Two cases are possible.
Case 1=>
\(N\) is a prime number.=> \(N\) will have 2 factors
Case 1->
\(N=2^2\)=> \(N\) will have 3 factors.

Hence Not sufficient.

Hence A.

120)If \(n=3*4*p\) where p is a prime number greater than 3,how many different positive non-prime divisors does n have, excluding 1 and n?
(A) Six
(B) Seven
(C) Eight
(D) Nine
(E) Ten
Show Spoiler
B.
\(n=3*2^2*p\)
n will therefore have 12 factors including1,n and its prime factors 2,3,p
So excluding them all n will have -> 12-5=7 factors.
Hence B

121)Data Sufficiency->How many divisors does the positive integer \(N\) have?
(1)\(27N^3\) has 16 factors.
(2)\(90<N^3<200\)
Show Spoiler
D.
In statement 1 we can collude that N is a prime number.
In statement 2 => N must be 5.
Hence D.

122)Data Sufficiency-> How many prime factors does positive integer n have?
(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.
Show Spoiler
E.
Taking test cases =>
n=5^3
n=5*3
We can choose E.

123)What is the greatest prime factor of \(2^{10}*5^4 - 2^{13}*5^2 + 2^{14}\)?
(A) 2
(B) 3
(C) 7
(D) 11
(E) 13
Show Spoiler
C.

124)Data Sufficiency->If p is a positive integer, is 2p + 1 prime?
(1) p is prime.
(2) units digit of p is not prime.
Show Spoiler
E.

125)What is the greatest prime factor of \(6^8−3^8\) ?
A) 3
B) 11
C) 17
D) 19
E) 31
Show Spoiler
C.
Proper in action=> \(A^2-B^2=(A+B)*(A-B)\)

126)Data Sufficiency->If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of factors.
(2) \(p-1\) is a factor of \(p\).
Show Spoiler
B.
We need to check if p is prime or not.
Statement 1->B.
Lets use some test cases =>
p=2
p+1=3
Both have 2 factors as both are prime.
Hence p is prime
But wait.Lets find some other test case for which p is non prime.
p=21
p+1=22
Both have 4 factors
And p is clearly non prime.
Another case be 14 and 15
Hence not sufficient.
Actually there exist ∞ such cases.
Statement 2->
For every positive integer >1 p and p-1 are always co-prime.
The only value of p possible is 2.
Hence p=2,which is a prime number.
Hene B

127)Data Sufficiency->If p is a positive integer, is integer k prime?
(1) 3p + 3 = k
(2) 7! + 3 = k
Show Spoiler
D.

128)Data Sufficiency->k is a positive integer. Is k prime?
(1) At least one number in the set {1, k, k + 7} is prime.
(2) k is odd.
Show Spoiler
C.
From statement 1->Using test cases=> k =4 or k=3
Hence not sufficient.
From statement 2=> k=3 or k=15
Hence not sufficient
Combing them => k+7 will be an even number >2, so it cannot be prime.
Hence k must be prime so that one out of 1,k,k+7 is prime.
Hence C.

129)Data Sufficiency->What is the value of the integer p?
(1) p is a prime number.
(2) 88 ≤ p ≤ 95
Show Spoiler
C.
Note->Only prime value in the boundary mentioned is 89.

130)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2)900 ≤ n < 1100
Show Spoiler
E.
Note-> For combination statement=>
Two examples for n=31^2=> n has less than 4 factors
or n=900=> n has more than 4 factors

131)If x is an integer and x^2 is even, which of the following must be true?
I. x is odd.
II. x is even.
III. x^3 is odd.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
Show Spoiler
B.
Property in action=> x and x^n always have the same even/odd nature if n≠0

132)Data Sufficiency->If K is a positive 3-digit number, is K prime?
(1) The last digit of K is not even
(2) K is the smallest number possible where the hundreds digit is the sum of the tens and units digit. Tens and units digits are equal.
Show Spoiler
B.
Statement 1 is off course nor sufficient.
We can use test cases => 211 is prime and 145 is not prime.
Notice that from statement 2 => K must be 211.
And 211 is a prime number as it is not divisible by any primes less than or equal to the square root of 211.
Hence B.

133)Data Sufficiency->If x and y are positive integers, is y divisible by 3?
(1) y = 2x^3 + 9x^2 - 5x.
(2) x is an odd number
Show Spoiler
A.
It is all about patterns.
Here is what i did in this Question=>

We need to see if y/3 is an integer or not.
We are given that a and y are positive integers.
Statement 1=>
\(y = 2x^3 + 9x^2 - 5x\)
Taking out x as a factor we get -> \(x[2x^2+9x-5]=> x(2x-1)(x+5)\)
Now putting in the values of x=> y is always a multiple of 3.
Hence sufficient.

Statement 2=>
No clue of y=> Not sufficient.

Hence A.

134)If \(x\) and \(y\) are positive integer and \(xy\) is divisible by 4, which of the following must be true?

A) If \(x\) is even then \(y\) is odd.
B) If \(x\) is odd then \(y\) is a multiple of 4.
C) If \(x+y\) is odd then \(y/x\) is not an integer.
D) If \(x+y\) is even then \(x/y\) is an integer.
E) \(x^y\) is even.
Show Spoiler
B.

135)If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
Show Spoiler
B.
Source->Official Guide

136)Data Sufficiency->If x is a prime number, what is the value of x?
(1) 2x + 2 is the cube of a positive integer.
(2) The average of any x consecutive integers is an integer.
Show Spoiler
E.
We are told that x is a prime number and asked about its value.

Statement 1->
2x+2=t^3
for any positive integer t.
Lets use some test cases =>
2x+2=1
x=1/2=> Not allowed
2x+2=8
x=3=> Allowed
2x+2=27
x=25/2=> Not allowed
2x+2=64
x=31=>Allowed

Hence 3 and 31 are both acceptable values => Not sufficient.

Statement 2->
Remember for set of consecutive integers is an AP series.
Mean can be of the form p if x is odd (p is an integer )
Mean can be of the form p.5 if x is even (p is an integer)
RULE->Sum of n consecutive integers is always divisible by n for n being odd and never divisible by n for n being even.
Hence this statement tells us that x is odd.
But their are ∞ odd primes => Not sufficient.
Combing them x=3 and x=31 are both acceptable values.
Hence not sufficient .

Hence E.

137)The sum of prime numbers that are greater than 60 but less than 70 is
(A) 67
(B) 128
(C) 191
(D) 197
(E) 260
Show Spoiler
B.
Only 2 prime are present in the boundary -> (60,70) => 61 and 67.
Sum =128.
Hence B.
Source->Official Guide

138)Data Sufficiency->If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?
(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84.
Show Spoiler
B.
Given data -> x,y,z are positive integers.
We are asked about the Greatest prime factor of x*y*z

Statement 1->
GCD(x,y,z)=7
Lets use test cases->
Case 1=>
7
7
7
GCD=> 7
And Greatest prime =7
Case 2=>
7
7
7*13
GCD=>7
Greatest prime factor is 7*13

Actually this statement tells us that the greatest prime factor must be greater than or equal to 7.
Hence not sufficient.

Statement 2->
LCM(x,y,z)=84=2^2*3*7
Now if x,y,z had any prime factor other than 2,3,7 => It must have been present in the LCM.
In LCM => We pick the greatest power of each Prime factor.
Hence the greatest prime in the Product must be 7.
Hence sufficient.
Hence B.


139)If n=6p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?
A)2
B)3
C)4
D)6
E)cannot be determined.

Show Spoiler
E.
Two cases exist here=>
Case 1-> p=3=> Even divisors ->3
Case 2-> p=prime >3 => even divisors ->4
Hence E.

140)Given that N=a^3*b^4*c^5 where a, b and c are distinct prime numbers, what is the smallest number with which N should be multiplied such that it becomes a perfect square, a perfect cube as well as a perfect fifth power?

A. a^3*b^4*c^5
B. a^5*b^4*c^3
C. a^2*b^3*c^5
D. a^7*b^6*c^5
E. a^27*b^26*c^25
Show Spoiler
E.
Perfect square => Powers of all primes=> Multiple of 2.
Perfect cube=>Powers of all the primes => Multiple of 3.
Perfect fifth power => Powers of all the primes => Multiple of 5

Hence powers of all the primes must be a multiple of all->2,3,5 => LCM=30
Smallest multiplication => \(a^{27}*b^{26}*c^{25}\)
Hence E.

141)Data Sufficiency->Is the integer b divisible by 6 ?
(1) 8b is divisible by 3.
(2) 9b is divisible by 12.

Show Spoiler
C.
We need to see if B is divisible by 6 or not.
Statement 1=> From this statement we can conclude that B must be divisible by 3.
E.g-> 3,6
Not sufficient.
Statement 2=> Form this statement , we can conclude that B must be divisible by 4
E.g->4,4*6
Not sufficient.
Combining the two statement => B must be divisible by 12.
Hence it must be divisible by 6 too.

Hence C.

142)Data Sufficiency->If k is a positive integer, is k a prime number?
(1) No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly.
(2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
Show Spoiler
D.
In order to check whether a given number is prime or not =>We must check its divisibility with all the prime numbers less than or equal to the square root of that number.
So clearly A is sufficient.

Statement 2 is a bit a trickier.
Lets compare the values of √k and k/2 for integers k>5
10=> √10=3.something 10/2=5
100=> √100=10 100/2=50
10000=> √10000=100 10000/2=5000
Clearly as the number is increasing the gap between √k and k/2 is also increasing.
Hence k must not have any prime factors between 1 and √k too.[/highlight]

Thus k is a prime number.
Hence Sufficient.

Hence D.


143)The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example , the prime sum of 12 is 7, since 12 = 2 x 2 x 3 and 2 +2 + 3 = 7. For which of the following integers is the prime sum greater than 35 ?
(A) 440
(b) 512
(C) 620
(D) 700
(E) 750
Show Spoiler
C.
Source->Official-Guide


144)Data Sufficiency->If a and b are integers, is a + b divisible by 15?
(1) a + b is divisible by 30.
(2) a + b is divisible by 5
Show Spoiler
A.

145)Data Sufficiency->If A and B are positive integers, is B divisible by A?
(1) 2B/A is an integer.
(2) B^2/A is an integer
Show Spoiler
E.
We are told that A and B are positive integers and we are asked if if B/A is an integer or not.
Lets use some test cases here.

Statement 1->
2B/A is an integer.
Let A=1 and B=1 => 2B/A is an integer and B/A =1 which is an integer.
Let A=2 and B=1 => 2B/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.

Statement 2->
B^2/A is an integer.
Let A=1 and B=1 => B^2/A is an integer and B/A=1 which is an integer.
Let A=4 and B=2 => B^2/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.

Combing the two statements ->
Let A=1 and B=1 => B/A is an integer.
Let A=4 and B=2 => B/A is not an integer.
Hence not sufficient.
Hence E.

146)Data Sufficiency->If n is a positive integer is n-1 divisible by 3 ?
(1) n^2+n is not divisible by 6.
(2) 3n=3k+3 where k is a positive multiple of 3
Show Spoiler
D.

147)Data Sufficiency->If n is a positive integer, is n – 1 divisible by 3?
(1) n^2 + n is not divisible by 6.
(2) 3n=k+3 where k is a positive multiple of 3.
Show Spoiler
A.

148)Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, …, x + 59}?
A)0
B)17
C)18
D)23
E)24
Show Spoiler
A.

149)The smallest prime factor of 899 is \(x\). Which of the following is true of \(x\)?
A. \(1 \lt x \le 7\)
B. \(7 \lt x \le 14\)
C. \(14 \lt x \le 21\)
D. \(21 \lt x \le 28\)
E. \(28 \lt x \le 35\)
Show Spoiler
E.
Notice 899=900-1=> 30^2-1=> 29*31
Both 29 and 31 are primes.
Hence smallest prime factor -> 29.
Hence E.

150)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2) 150 ≤ n < 200
Show Spoiler
E.
We need to see if the number of factors of n is greater than or equal to 4 or not.

A few key links =>

1 is the only number that has one factor.

A prime number has exactly two factors.

For any number to have 3 factors it must be of the form => Prime^2 i.e square of a prime number.

Lets get on with the statements=>

Statement 1->
n is not prime.
Lets play around with some test cases.
n=1 => One factor => n has less than 4 factors
n=200=> 2^3*5^2 => 12 factors => n has more than 4 factors.
Hence insufficient.

Statement 2->
n lies in the range [150,200]
Again lets use some test cases here.
n=151=> Prime number as it is not divisible by any prime number less than equal to the square root of 151.
So n has 2 factors which is less than 4.
n=200=> 12 factors => more than 4 factors.
Not sufficient.

Combing the two statements =>
n is not prime.
ans n≠1
So n can never has one or two factors.

But what about three?
Can n be square of a prime ?
YES for n=13^2=169=> n has 3 factors which is less than 4
For all values of n => n will have more than 3 factors.
Hence not sufficient.
Hence E.

151)Data Sufficiency->If p is a positive integer, what is the value of p?
(1) p/4 is a prime number.
(2) p is divisible by 3
Show Spoiler
C.
Combing the two statements => p must be 12.
Source->Gmat-Prep

152)Data Sufficiency->If a and b are integers, is (ab+2)(ab+3)(ab+4) divisible by 12?
(1) a=even
(2) b=odd
Show Spoiler
A.

153)An integer n that is greater than 1 is said to be "prime-saturated" if it has no prime factor greater than or equal to √n.
Which of the following integers is prime saturated?
A) 6
B) 35
C) 46
D) 66
E) 75
Show Spoiler
E.
Source->Gmat-Prep

154)Data Sufficiency->If p is a positive integer, is p^2 divisible by 96?
(1) p is a multiple of 8.
(2) p^2 is a multiple of 12
Show Spoiler
C.

155)If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?
(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2
Show Spoiler
B.
Plugging in x=5 => Sum =75
Only option that matches is B.
Hence B.
Note=> While plugging in numbers we must check for each option.There are cases when you might get more than one answer.
In such cases we must alter our test cases to choose between the remaining options.

156)Data Sufficiency->Does p^2 = q if p is a prime number?
(1) q^2 – p^2 =0
(2) p^2 = 49
Show Spoiler
A.


157)A number is said to be prime saturated if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two digit prime saturated integer ?
A. 99
B. 98
C. 97
D. 96
E. 95
Show Spoiler
D.
Source-> GMAT-prep

158)Data Sufficiency->How many different prime factors does x have?
1) \(5x^2\) has two different prime factors
2) \(x > 1\)
Show Spoiler
E.
Key Properties applicable in this question =>
X and X^n always have the exact same prime factors.

When we multiple a positive integer x with some other positive integer n => The number of prime factors of nx may be greater than or equal to the number of prime factors of x.
We need the number of prime factors of x.
Statement 1->
5x^2 has primes
x can have 2 primes (if 5 is one of its primes)
x can have 1 prime (if 5 is not one of its primes)
Hence not sufficient.
Statement 2->
x>1
Clearly not sufficient.
Combing the two statements =>
Still not sufficient.

E.g =>
x=3
or
x=5*3
Satisfy both the equations.

Hence E.

159)How many different prime factors does positive integer x have?
(1) \(1 < x < 6\)
(2) \(5x^2\) has four factors.
Show Spoiler
D.
We need the number of prime factors of the positive integer x.
Statement 1=>
x=>(1,6)=> x=2,3,4,5=> In each case => x will have just one prime factor=> Sufficient.
Statement 2=> The only value of x possible is 5.
Hence x will have only one factor. Sufficient

Hence D.

160)Data Sufficiency->If N is a positive integer, does N have exactly three factors?
(1) The integer N^2 has exactly five factors
(2) Only one factor of N is a prime number
Show Spoiler
A.
We need to check if n has 3 factors or not.

Firstly a few key kicks =>
1 is the only number that has one factor.
A prime has exactly 2 factors.
For a number to have 3 factors =>It must be of the form => Prime^2 i.e square of a prime number.

Lets jump into Statements.
Statement 1->
n^2 has 5 factors.
This can only happen if n=Prime^2
Hence sufficient.

Statement 2->
This statement is basically saying that n has only prime factor.
But it does not tell us anything about the exponent of that prime.
E.g=> n=2^2=> 3 factors
n=2^100=>101 factors
Hence not sufficient.
Hence A.

161)Data Sufficiency->What is the value of x?
(1) x is the square of an integer.
(2) 577 < x < 675
Show Spoiler
C.

162)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 4 prime numbers between 11 and x
2 )There is no y such that 1<y<x and y is the divisor of x.
Show Spoiler
A.

163)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number
Show Spoiler
B.

164)For how many positive integers is the number of positive divisors equal to the number itself ?
A) none
B) one
C) two
D)three
E)cannot be determined
Show Spoiler
C.
One and two are only number for which the number of factors is equal to the number itself.
Hence C.

165) The positive integer 2000 has how many factors?
A)2
B)10
C)12
D)16
E)20
Show Spoiler
E.





166)Data Sufficiency->If x is a perfect square greater than 1, what is the value of x?
(1) x has exactly 3 distinct factors.
(2) x has exactly one positive odd factor
Show Spoiler
C.

We are given that x is a perfect square and are asked its value.
Statement 1->
x has 3 factors.
Some quick facts on factors ->
1 is the only number that has 1 factor.
A prime as 2 factors.
A number of the form Prime^2 has 3 factors.
Hence x must be of the form Prime^2
E.g=> 2^2 or 3^2 etc.
Hence not sufficient.

Statement->2
As 1 is the factor of every number => This statement tells us that x must be even.
E.g => 2^2 or 2^4 or 2^6
Basically it cannot have any prime factor other then one.
So it must be of the form 2^even number .
Hence not sufficient.

Combing the two statements=>
x must be 2^2 to have 4 factors.
Hence sufficient.
Hence C.

167)Data Sufficiency->How many factors does y have?
(1) y is the cube of an integer.
(2) y is the product of 2 distinct positive digits.
Show Spoiler
C.
We need to get the number of factors of y.

Statement 1->
y is a perfect cube.
y can be negative or zero or positive.
It would have been nice if the Question Stem had read-> y is a polite integer.
Anyways its clearly not possible to get the factors.
Not sufficient.

Statement 1->
From this statement we can infer that y must be positive.
y=2*3=> 6(4 factors)
y=1*5=>5(two factors)
y=3*9=>27=> 4 factors.
Hence not sufficient.

Combing the two statements=> The only values that y can take are 8 and 27 and they both have 4 factors.

Hence C.

168)Data Sufficiency->If x, y and k are integers, is xy divisible by 3?
(1) y = 2^(16) - 1
(2) The sum of the digits of x equals 6^k
Show Spoiler
A.
Note->k can be zero too.

169)Data Sufficiency->If x is a prime number, what is the value of x?
(1) x is less than 15
(2) (x-2) is multiple of 5
Show Spoiler
E.
Note-> x=2 and x=7 satisfy both the statements.

170)Data Sufficiency->How many prime factors does positive integer n have?
(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.
Show Spoiler
E.
Note-> Values such as x=7^2 and 7*3 will satisfy both the statements.
Hence E.

171)If x = 13y, where y is a prime number greater than 2, how many different positive even divisors does x have, including x?
A. 0
B. 1
C. 2
D. 3
E. It cannot be determined from the information given
Show Spoiler
A.

172)Data Sufficiency ->If b is a positive integer, then b has how many distinct positive factors?
(1) all of b's factors are also factors of 62.
(2) b is the product of one even prime number and one odd prime number.
Show Spoiler
B.
We need to get the number of factors of b.


Statement 1->
This is just another way of saying that 62/b will be an integer i.e b is a factor of 62.
b=2
b=1
b=62
Etc .
Hence not sufficient.

Statement 2=>
b=Product of two prime numbers that are different.
Hence b will be of the from prime1 *prime2
Hence number of factors will be 4.

Hence sufficient.
Hence B.

173)Data Sufficiency->How many different factors does the integer n have?
(1) n = (a^4)(b^3) where a and b are different positive prime numbers.
(2) The only positive prime numbers that are factors of n are 5 and 7.
Show Spoiler
A.
We need to get the number of factors of positive integer n.

Statement 1=>
As a and b are "different" prime numbers => Number of factors of a must be 5*4=20
Hence sufficient .
Statement 2=>
There exist ∞ numbers with the same set of prime numbers.
E.g
5*7=> Four factors.
5^2*7^2=> Nine factors.
Etc.
Hence not sufficient.

Hence A.

174)Data Sufficiency->If x and y are positive integers and x + y = 3^x, is y divisible by 6?
(1) x is odd.
(2) x is a multiple of 3.
Show Spoiler
Given info->x and y are positive integers.
x+y=3^x
In order for y to be divisible by 6 => It must be divisible by both 2 and 3.
Statement 1->
Test Case 1->
x is odd
x=1
y=y
x+y=3^1=> Yes
Clearly y is not divisible by 6.
Test Case 2->
x =3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6
Hence not sufficient.
Statement 1->
x is a multiple of 3.
So y must be a multiple of 3 too.
Now=>
Test case 1->
x=3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6.
Test case 2->
x=6
y=3^6-6=>Odd number -Even number =>Odd number => Never divisible by 6.
Hence not sufficient.
Combing the two statements=>
From Statement 1=> y=> even
From Statement 2=> y must be a multiple of 3.
Hence y must be a multiple of 6.
Hence Sufficient.
Hence C.

175)Data Sufficiency ->If x^3 – x = n and x is a positive integer greater than 1, is n divisible by 8?
(1) When 3x is divided by 2, there is a remainder.
(2) x = 4y + 1, where y is an integer.
Show Spoiler
D.
As n=x^3-x=> (x-1)*x*(x+1)=> If x is odd => both x-1 and x+1 will be consecutive even integers.So one of them would be multiple of 4 and other would be a multiple of 2 for sure.Hence n would be a multiple of 8.

Statement 1->
3x is odd => x is odd => Sufficient.
Statement 2->
x=4y+1 => Even+odd=odd
Hence Sufficient.

Hence D.


176)If a positive odd integer N has p positive factors, how many positive factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
Show Spoiler
B.

177)Data Sufficiency->Is the integer n odd?
(1) n is divisible by 3
(2) 2n is divisible by twice as many positive integers as n
Show Spoiler
B.
We are told that n is an integer.
Statement 1->
n/3=integer.
E.g=> n=3 or n=6 etc.
Hence not sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be 2 times the number of factors of n.
This is because any odd number will just have odd divisors.It cannot have any even divisor as 2 is not its prime factor.
After we multiple 2 to n => The numb of factors will get doubled as for every odd factor there would be an even factor.

Hence n is odd
Thus-> This statement is sufficient.
Hence B.

178)If x is a prime number greater than 5, y is a positive integer, and 5y=x^2+x, then y must be divisible by which of the following?

I. 5
II. 2x
III. x+1

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Show Spoiler
B.

179)If P is a prime number greater than 5, what is the remainder when P^2 is divided by 8.
A) 4
B) 3
C) 2
D) 1
E) Cannot be determined
Show Spoiler
D.
All prime numbers greater than 2 are odd.
Hence p must be of the form => 2k+1
p^2=> 4k^2+4k+1=> 4k(k+1)+1= 8k'+1
Hence it will always leave a remainder 1 with 8.

Another important takeaway ->
For any odd number k => k^2 will always leave a remainder 1 with 8.

180)Data Sufficiency->Is positive integer N divisible by 3?

(1) N^2/36 is an integer
(2) 144/N^2 is an integer
Show Spoiler
A.
A very useful property -> X and X^n always have the exact same prime factors.
For N to be divisible by 3 => 3 must be the prime factors of N.

Statement 1=>
N^2/36 is an integer.
N^2 has both 2 and 3 as it prime.
Thus N must have 3 as its prime too.
Hence Sufficient.
Statement 2=>
Lets use test Cases here.
N=1 => 144/1^2 =Integer => 1 is not divisible by 3.
N=3 => 144/N^2 =integer => 3 is divisible by 3.
Hence not sufficient.

Hence A.

181)Data Sufficiency->If x is an integer, is x^3 divisible by 7?
(1) 3*x^12 is divisible by 7
(2) 3*x^4 is divisible by 7
Show Spoiler
D.
Property in action =>X and X^n always have the exact same prime factors.

182)Data Sufficiency->How many different prime factors does positive integer n have?
(1) 48< n^2 < 99
(2) 125n^2 has twelve factors
Show Spoiler
D.

183)Data Sufficiency->Is the integer k divisible by 6?
(1) 6k is divisible by 12.
(2) 9k is divisible by 12.
Show Spoiler
E.
We can take k=4 and k=4*6 to arrive at E.

184)If n = 3^8 - 2^8, which of the following is NOT a factor of n?
(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
Show Spoiler
C.
Here => Using a^2-b^2=(a+b)*(a-b)=>3^8-2^8 = 97*13*5.
Clearly 35 is not a factor.
Hence C.
Source-> Official Guide.

185)Data Sufficiency->If n is a positive integer, is n^3 – n divisible by 24 ?
(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6
Show Spoiler
A.
Firstly n^3-n= (n-1)n(n+1)
It is a product of 3 consecutive integers.
Regardless of what the value of n is => It will always be a multiple of 6.
Reason -> Product of n consecutive integers is divisible by n!

Now lets see two cases ->

Case 1-> n is odd
If n is odd => n-1 will be even and n+1 will be even too.
Further more one out of them will be multiple of 4.
Thus (n-1)n(n+1) will be multiple of 8.
As it is already a multiple of 6 => we can conclude that (n-1)n(n+1) will be a multiple of 24.

Case 2-> n is even
In that scenario it is not possible for us to conclude anything except (n-1)n(n+1) will be multiple of 6.



Hence we just need to see if n is odd or not.

Statement 1->
n=2k+1 for some integer k.
Hence n=even+odd=odd
Hence n must be a multiple of 24.
Hence sufficient.

Statement 2->
Lest use some test cases=>
n=2 => (n-1)n(n+1) = 1*2*3 => Not a multiple of 4.
n=3 => (n-1)n(n+1)=2*3*4 =>Multiple of 24.

Hence not sufficient.

Hence A.
Source->Official Guide.


186)For which of the following values of n is (100+n)/n NOT an integer?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Show Spoiler
C.
Source->Official Guide.


187)Data Sufficiency->Is a*b*c divisible by 24?
(1) a, b, and c are consecutive even integers
(2) a*b is divisible by 12
Show Spoiler
A.
We need to see if a*b*c is divisible by 24 or not.


Statement 1->
a,b,c are consecutive evens =>Let the consecutive evens be =>
2n
2n+2
2n+4

Taking the product => 2n(2n+2)(2n+4)=> 8n(n+1)(n+2)

Product of t consecutive integers is always divisible by t!
Hence 8n(n+1)(n+2)=8*6k = 48k for some integer k.
Clearly it will be divisible by 24.
Hence sufficient.

Statement 2->
Here we have no clue whether a,b,c are integers or non integers.
Hence not sufficient.
Hence A.

188)Data Sufficiency->If x is the product of integers a, b, c, and d, is x divisible by 128?
(1) a = 24
(2) a, b, c, and d are consecutive even integers
Show Spoiler
B.
We are told that a,b,c,d are all integers.
We need to see if x=a*b*c*d is divisible by 128 or not.

Statement 1->
Lets use test cases.
Case 1=>
a=24
b=1
c=1
d=1
x=24 => Clearly it is not divisible by 128.
Case 2=>
a=24
b=1
c=1
d=128
x=24*128 => divisible by 128.
Hence not sufficient.

Statement 2->
Consecutive evens =>
a=2n
b=2n+2
c=2n+4
d=2n+6


Taking the Product => x=2n(2n+2)(2n+4)(2n+6)=> 8*n(n+1)(n+2)(n+3)

Property -> Product of n consecutive integers is always divisible by n!

Hence x=8*24k For some integer k=> 128k'
Hence x must be divisible by 128.

Hence sufficient.

Hence B.

189)Data Sufficiency->How many positive factors does positive integer N have?
1) N^3 has Four positive factors
2) 5N has four positive factors
Show Spoiler
A.
From statement 1-> N must be a prime number.Hence it will always have 2 factors.Hence sufficient.
From statement 2 -> Two cases are possible => N=5^2 and N=Any prime number other than 5.
Hence N can have 3 or two factors.
Hence not sufficient.

Hence A.

190)Which of the following statements must be true?
A)If n is even then n^3-n is divisible by 24.
B)The sum of first 1000 prime numbers is even.
C)The product of first 32 primes is odd.
D)2 and 3 are the only consecutive prime numbers.
E)If n is odd then n^3-n is always divisible by 24.
Show Spoiler
E.
Mote-> D is not true as there is a BIG difference between consecutive prime numbers and consecutive numbers that are prime.
So 5,7 or 11,13 or 17,19 etc are all consecutive prime numbers.

2,3 are the only consecutive integers that are also prime numbers.

191)Which of the following is NOT a factor of 10! ?
A. 1440
B. 625
C. 160
D. 80
E. 50
Show Spoiler
B.
Number of 5's in 10!=2
So 625 which has five 5's can never be a factor of 10!
Hence B.

192)How many prime factors does the positive integer N have?
A)N^3 has 5 prime factors.
B)10N^3 has 7 Prime factors.
Show Spoiler
A.

193)Data Sufficiency->How many prime factors does N have?
(1) N is a factor of 7200.
(2) 180 is a factor of N.
Show Spoiler
C.
We need to get the number of prime factors of N.

Statement 1->
N is a factor of 7200
7200=2^5*3^2*5
Hence N can have -->
Zero prime factor if N=1
One prime factor
Two prime factors
Or Three Prime factors.
Hence not sufficient.


Statement 2->
180 is a factor of N.
Hence N=2^2*3^2*5*k for some integer K.
Hene N must have 2,3,5 as its prime factors.
Other than that N can have any Prime factors ≥3
Hence not sufficient.

Combining the two statements =>
N must have exactly three prime factors>
Hence sufficient.


Hence C.

194)Data Sufficiency->If F is the prime factorization of N!, how many factors in F have an exponent of 1?
(1) 30 ≤ N ≤ 40
(2) 25 ≤ N ≤ 35
Show Spoiler
E.

195)Data Sufficiency->Given that \(x\) is an integer and \(x\) is positive, is \(\frac{210}{x}\) also an integer??
(1) \(x\) is a prime number.
(2) \(x\) < 8
Show Spoiler
C.

Observe that 210=2*3*5*7
We need to see if 210/x is an integer or not.
Statement 1->
x is prime
E.g=> x=13 => No
x=7 => Yes
Hence not sufficient.


Statement 2->
x<8
E.g-> x=4 -> No
x=7 => Yes
Hence not sufficient.
combining the two statements =>
x can be 2,3,5,7
Each one if them is a factor of 210.
Hence 210/x will always be an integer.
Hence sufficient.
Hence C.

196)Which of the following is NOT a factor of the product of the first 50 positive multiples of 4 ?
A. \(17^2\)
B. \(11^4\)
C. \(7^6\)
D. \(47^{12}\)
E. \(2^{124}\)
Show Spoiler
D.
Firstly the product of first positive 50 multiples of 4=>
4*1*4*2*4*3*4*4*4*5.....4*50=> 4^50*50!

Secondly notice Option D is 47^2 and 47 is a prime number.

Number of 47's in 4^50*50!=> one

Hence 47^2 will never be its factor.

Hence D.

197)Data Sufficiency->Is the integer \(n\) odd?
(1) a is an integer and \(n=a^7+a^5+a^3+a^2+2a+1\)
(2) 2n is divisible by twice as many positive integers as n
Show Spoiler
D.
Statement 1->
Here a can be even or odd => n will always be odd.
Hence sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be twice the number of factors of n.
Hence n is odd => Sufficient.

Hence D.

198)If a positive integer n has 211 factors,then how many prime factors does n have?
A)one
B)two
C)three
D)Four
E)cannot be determined
Show Spoiler
A.
Notice here that 211 is a prime number as it is not divisile by any prime factor less than or equal to the square root of 211.
Therefore the number n can only be of the form => Prime^210
Hence it will always have one prime factor.

If the number of factors of any integer is a prime number => The integer will always have a single prime factor.
Hence A

199)If a positive integer N has p factors ; how many factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
Show Spoiler
E.

200)Data Sufficiency ->Is integer k a prime number?
(1) k = 10! + m, where 1 < m < 8
(2) k is a multiple of 7
Show Spoiler
A.


Note-->PM me if you find any answers/solutions inaccurate/inadequate.

Any Feedback would be Appreciated.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  26 Nov 2016, 09:01
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Day 4



Mock 4



Topic Covered->Units Digits & Divisibility.

Number of Questions --> 70.




1)If |x+3|=5,then what are the possible values of x?
Show Spoiler
-8 and 2.

2)If x is a number such that –2 ≤ x ≤ 2, which of the following has the largest possible absolute value?
A. 3x – 1
B. x^2 + 1
C. 3 – x
D. x – 3
E. x^2 – x
Show Spoiler
A.
We need the maximum absolute value.
So actually we just need the maximum magnitude as the sign won't matter.
Option 1->Maximum at x=-2 => Value=-7 =>Magnitude =7
Option 2->Maximum at x=-2 or 2=> Value=5 =>Magnitude =5
Option 3->Maximum at x=-2=>Value=5=>Magnitude =5
Option 4->Maximum at x=-2=>Value=-5=>Magnitude=5
Option 5->Maximum at x=-2=>Value=6=>Magnitude=6
Hence A.

3)For what values of x is the expression |\(x^2-3\)| minimum?
Show Spoiler
\(-√3\) and \(√3.\)

4)If x is an integer and the units digit of x + x^2 + x^3 is odd, which of the following must be even?
A. x
B. x^2
C. x^3
D. 3x
E. x + x^2
Show Spoiler
E.

5)What is the units digit of \(789+4123+2666+842358\) ?
Show Spoiler
6.

6)What is the units digit of \(1999*31452*600003*71456746\) ?
Show Spoiler
4.

7)What is the units digit of \(23^{99}*14^{352}+9002^{1003}*918^{437986}\) ?
A)1
B)2
C)3
D)4
E)5
Show Spoiler
D.

8)What is the remainder when \(104^{358}+7^{29}\) is divided by 5?
A)4
B)3
C)2
D)1
E)0
Show Spoiler
B.

9)If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

A. I​ only
B. III only
C. II and III
D. I and II
E. I​, II, and III
Show Spoiler
E.

10)If 10 and 8 are factors of Q, which of the following is the largest number that must divide evenly into Q?
A. 12
B. 40
C. 50
D. 75
E. 100
Show Spoiler
B.

11)Find the rightmost non-zero digit of the number \(345637300^{3725}\)?
A)1
B)3
C)5
D)7
E)9
Show Spoiler
B.

12)If x = \(3^{21}\) and y = \(6^{55}\), what is the remainder when x*y is divided by 5?
A)0
B)1
C)2
D)3
E)4
Show Spoiler
D.

13)If x = \(3^{21}\) and y = \(6^{55}\), what is the remainder when 2x+y is divided by 5?
A)0
B)1
C)2
D)3
E)4
Show Spoiler
C.

14)If p is a positive integer, what is the units digit of Z, if Z = \(104^{4p + 1} * 277^{p + 1} * 93^{p + 2} * 309^{6p}\)?
A)0
B)2
C)4
D)6
E)8
Show Spoiler
B.

15)Data Sufficiency->If \(p\) and \(q\) are positive integers and \(X = 6^p + 7^{q+23}\), what is the units digit of \(X\)?
(1) \(q = 2p – 11\)
(2) \(q^2 – 10q + 9 = 0\)
Show Spoiler
B.

16)Data Sufficiency->The number \(x\) is a positive odd integer. If the unit digit of \(x^3\) is subtracted from the unit digit of \(x^2\), it results in 0. What is the unit digit of the number \(x + 7\)?
(1) The unit digit of the product of 105 and \(x\) is 5.
(2) When x is divided by 5, it leaves no remainder.
Show Spoiler
B.

17)Data Sufficiency->If M and N are positive integers greater than 1, what is the remainder when the expression \(22^{3M} * 39^{2N} + 14^{2(M+N)}\) is divided by 5?
(1) M = 13
(2) N = 14
Show Spoiler
A.

18)If Z = 1^1 * 2^2 * 3^3*...* 8^8, what is the remainder when Z is divided by 10?
A)0
B)1
C)2
D)3
E)4
Show Spoiler
A.

19)If Z = 1^1 + 2^2 + 3^3 +...+ 8^8, what is the remainder when Z is divided by 10?
A)0
B)2
C)4
D)6
E)8
Show Spoiler
E.

20)What is the units digit of \(31467^{32} * 97645^{23} * (32168^5 + 8652)^{479}\) ?
A)0
B)1
C)2
D)5
E)8
Show Spoiler
A.

21)What will be the units digit, when the product of the first 10 natural numbers is divided by 100?
A)0
B)2
C)4
D)6
E)8
Show Spoiler
E.

22)Find the product of 57 x 61 x 39 x 53.
A)7086751
B)7086953
C)7186959
D)7186965
E)7286977
Show Spoiler
C.

23)Find the units digit of the product of all the prime numbers between 1 and 13^13.
A)0
B)2
C)3
D)5
E)6
Show Spoiler
A.

24)What is the rightmost non-zero digit of \(90^{42}\)?
A)9
B)7
C)5
D)3
E)1
Show Spoiler
E.

25)Data Sufficiency->Find the units digit of \(5^{3n} + 9^{5m}\) , where m and n are positive integers.
(1)m is an odd integer.
(2)n is an even integer.
Show Spoiler
A.

26)If a is a positive integer, and if the units digit of \(a^2\) is 9 and the units digit of \((a+1)^2\) is 4, what is the units digit of \((a+2)^2\)?
A)1
B)3
C)5
D)6
C)14
Show Spoiler
A.
Source->Gmat-prep.

27)If x is a positive integer, what is the units digit of \((24)^{(2x+1)}(33)^{(x+1)}(17)^{(x+2)}(9)^{(2x)}\)
A)4
B)6
C)7
D)8
E)9
Show Spoiler
D.

28)If the units digit of x^3 is 6, what is the units digit of integer x?
A. 2
B. 3
C. 4
D. 6
E. 12
Show Spoiler
D.

29)What is the unit's digit of \(7^{75} + 6\)
A. 1
B. 3
C. 5
D. 7
E. 9
Show Spoiler
E.

30)What is the units digit of (5!*4! + 6!*5!)/31?
A. 4
B. 3
C. 2
D. 1
E. 0
Show Spoiler
E.

31)What is the units digit of \(6^{15} - 7^4 - 9^3\)?
A)8
B)7
C)6
D)5
E)4
Show Spoiler
C.

32)What is the units digit of 3^67?
A. 1
B. 3
C. 5
D. 7
E. 9
Show Spoiler
D.

33)What is the units digit of 18! + 4!?
A. 0
B. 1
C. 2
D. 3
E. 4
Show Spoiler
E.

34)What is the units digit of 17^381?
A. 1
B. 3
C. 5
D. 7
E. 9
Show Spoiler
D.

35)What is the units digit of 26! + 50! + 4! + 4!?
A. 8
B. 6
C. 4
D. 2
E. 0
Show Spoiler
A.

36)What is the units digit of (3^11)(4^13)?
A. 2
B. 4
C. 6
D. 7
E. 8
Show Spoiler
E.

37)What is the units digit of the expression 14^7−18^4?
(A)0
(B)3
(C)4
(D)6
(E)8
Show Spoiler
E.

38)What is the units' digit of the following expression (13)^5*(15)^4*(17)^5?
A. 0
B. 1
C. 3
D. 5
E. 9
Show Spoiler
D.

39)What is the unit’s digit of \(17^7\)?
A) 1
B) 3
C) 5
D) 7
E) 9
Show Spoiler
B.

40)What is the units digit of \(13^{27}\)?
A. 1
B. 2
C. 3
D. 7
E. 9
Show Spoiler
D.

41)What is the units digit of 18^47 ?
A. 0
B. 2
C. 4
D. 6
E. 8
Show Spoiler
B.

42)What is the units digit of (23^6)(17^3)(61^9)?
A. 1
B. 3
C. 5
D. 7
E. 9
Show Spoiler
D.

43)If x is a positive integer, what is the units digit of \(24^{5 + 2x}*36^6*17^3\)?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 8
Show Spoiler
A.

44)What is the units digit of 2222^(333)*3333^(222)?
A. 0
B. 2
C. 4
D. 6
E. 8
Show Spoiler
E.

45)What is the units digit of \(6^m * (2^7+1)\) for a positive integer m?
A. 1
B. 3
C. 4
D. 7
E. 0
Show Spoiler
C.

46)Data Sufficiency->If z is an integer, is z even?
(1) z/2 is not an odd integer.
(2) z + 5 is an odd integer.
Show Spoiler
B.
We need the even odd nature of integer z

Statement 1->
z=5
z=4

Taking these two test cases we can say that this is an insufficient statement

Statement 2->
z-odd=odd => z=> odd+odd=even
Hence sufficient

Hence B.

47)Data Sufficiency->If z is an integer, is z even?
(1) (z + 7)/3 = 2m + 1, where m is an integer.
(2) z^3 is even.
Show Spoiler
D.

48)Data Sufficiency->Is z an even integer?
(1) z/2 is an even integer.
(2) 3z is an even integer.
Show Spoiler
A.
Note ->Statement B is not sufficient as we are not told whether z is an integer or not.

49)Data Sufficiency->If z is an integer, is z even?
(1) z/2 is even.
(2) 3z is even.
Show Spoiler
D.

50)Data Sufficiency->The product of integers x, y, and z is even, is z even?
(1) x/y = z
(2) z = xy
Show Spoiler
B.

51)Data Sufficiency->The product of integers x, y, and z is even, is x even?
(1) x/y = z
(2) z = xy
Show Spoiler
A.

52)Data Sufficiency->If, x, y, and z are integers, is x even?
(1) 10^x = 4^y*5^z
(2) 3^(x + 5) = 27^(y + 1)
Show Spoiler
A.

53)Data Sufficiency->If x, y, and z are all positive integers, is x + y + z even?
1) \(\frac{2x}{(y+z)}\) = odd.
2) \(\frac{x^2}{4yz}\) is an integer.
Show Spoiler
C.

54)Data Sufficiency->If x,y,z and t are integers,is \(x + y – z + t\) even?
(1)\(x + y + t\) is even.
(2)\(t*z\) is odd.
Show Spoiler
C.

55)If x is an integer, which of the following must be an even integer?
A: x^2-x-1
B: x^2-4x+6
C: x^2-5x+5
D: x^2+3x+8
E: x^2+2x+10
Show Spoiler
D.

56)Data Sufficiency->Is a even?
(1) 2a is even.
(2) \(\sqrt{a}\) is even.
Show Spoiler
B.

57)Data Sufficiency->Is n an even?
1) 3n=even
2) 5n=even
Show Spoiler
C.

58)Data Sufficiency-> Is the integer a even?
(1) a is divisible by 7.
(2) a is divisible by 14.
Show Spoiler
B.

59)Data Sufficiency->Is n an even?
1) 7n is even.
2) 28n is even.
Show Spoiler
E.

60)Data Sufficiency->If n is an integer, is n - 1 even?
(1) n - 2 is odd.
(2) n + 1 is even.
Show Spoiler
D.

61)Data Sufficiency-> Is N divisible by 4?
(1)N is a product of two even integers.
(2)N is a product of three consecutive integers.
Show Spoiler
A.

62)Data Sufficiency->Is positive integer n is divisible by 4 ?
1) \(n^2\) is divisible by 8.
2)\(\sqrt{n}\) is even integer.
Show Spoiler
D.

63)Data Sufficiency->If n is a positive integer, is n^3 - n divisible by 4?
(1) n = 2k+1, where k is an integer.
(2) n^2 + n is divisible by 6.
Show Spoiler
A.
Source->Official Guide.

64)Data Sufficiency->If n is an integer, is n even?
(1) 2n is divisible by 4.
(2) n^2 is even.
Show Spoiler
D.
Source-> Gmat-prep

65)Data Sufficiency->Is n an even number?
(1) n^2 = n
(2) n^3 = n
Show Spoiler
E.
We are asked if n is an even integer or not.
Statement 1->
n^2=n => n can be 0 or 1
0=even
1=odd
Hence not sufficient.
Statement 2->
n^3=n
n=0 or 1 or -1
Hence not sufficient.
Combing the two statements =>
n can be zero or one.
zero is even
one is odd.

Hence not sufficient.

Hence E.

66)Data Sufficiency->If n is an integer, is n even?
(1) n^2 - 1 is an odd integer.
(2) 3n + 4 is an even integer.
Show Spoiler
D.
Source->Official Guide.

67)Data Sufficiency->If n and m are positive integers, is n + m even?
(1) 5nm is odd.
(2) n – m is even.
Show Spoiler
D.

68)Data Sufficiency->If n and m are positive integers, is m a factor of n?
(1) n = 5(3^k), for any positive integer k.
(2) m = 3^(k-1), for any positive integer k.
Show Spoiler
C.

69)If x is an integer, which of the following must be an odd integer?
A)x^2-x-12
B)x^2-4x+6
C)x^2-5x+5
D)x^2+3x+8
E)x^2+2x+10
Show Spoiler
C.

70)If x is an even number,then which of the following statements must be divisible by 8?
A)3x^2
B)6x
C)4x
D)10x
E)x^3/2
Show Spoiler
C.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  26 Nov 2016, 09:18
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stonecold wrote:
If the GCD of m and 25 is 5 and GCD of m and 12 is 3,which of these can never be the value of positive integer m?

A)15
B)45
C)60
D)105
E)165


Test Using options -

A. \(15 = 3*5\), So, GCD ( 15, 25) is 5 & GCD ( 15, 12) is 3
B. \(45 = 3^2*5\), So, GCD ( 45, 25) is 5 & GCD ( 45, 12) is 3
C. \(60 = 2^2*3*5\), So, GCD ( 60, 25) is 5 & GCD ( 25, 12) is 1

We, need to go no further... Answer will be (C)..

A quick check will be Using the options.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  09 Dec 2016, 21:05
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Word Problem Set based on Fractions, Ratio,Interest and Percents -->


Q1)Class B has 50% more students than class A. Number of girls in class A is equal to number of boys in class B. The percentage of girls is the same in both classes. What percentage of the student group are boys?
A. 25 %
B. 33 %
C. 40 %
D. 50 %
E. 60 %
Show Spoiler
C.



Q2)The Bryant's flower shop situated at the New Plaza complex stocks four types of flowers. There are 1/3 as many violets as carnations, and 1/2 as many tulips as violets. If there are equal no. of roses and tulips, what percent of the flowers in the shop are carnations?

(A) 6.25

(B) 20

(C) 33

(D) 50

(E) 60
Show Spoiler
E.


Q3)If m = 9/25, w = 15/32, and m + w + c = 1, which of the following gives the values of m, w, and c in increasing order?

A. c, m, w
B. c, w, m
C. m, w, c
D. w, c, m
E. w, m, c

Show Spoiler
A.


Q4)XYZ Corporation has a ratio of 1 : 6 female to male. If the average salary for the female employees is $217,800 and XYZ pays $2,395,800 in salary to female employees, how many men work at XYZ Corporation?

A. 77
B. 66
C. 55
D. 24
E. 11
Show Spoiler
B.


Q5)Two libraries are planning to combine a portion of their collections in one new space. One third of the books from Library A will be housed in the new space along with One fourth of the books from Library B. If there are twice as man books in Library B as in Library A, what proportion of the books in the new space will have come from Library A?

A)1/3
B)2/5
C)1/2
D)3/5
E)7/12
Show Spoiler
B.


Q6)Question 7) Lisa spends 3/8th of her salary on rent and 5/12 on food. Her roommate, Carrie earns about twice as much as Lisa, spends 1/4th of her salary on the rent and 1/2 on food. If the two women decide to contribute the rest of their salary to charity every month, what fraction of their combined monthly income will they donate.

No options for this one, just tell me the fraction value :)
Show Spoiler
17/72


Q7)Dara ran on a treadmill that had a readout indicating the time remaining in her exercise session. When the readout indicated 24 min 18 sec, she had completed 10% of her exercise session. The readout indicated which of the following when she had completed 40% of her exercise session?

A. 10 min 48 sec
B. 14 min 52 sec
C. 14 min 58 sec
D. 16 min 6 sec
E. 16 min 12 sec
Show Spoiler
E.


Q8)Question 8) A restaurant spends one quarter of its monthly budget for rent and half of the rest for food and beverages. What percentage of the budget does the restaurant spend for food and beverages?

A. 23.5%
B. 32.5%
C. 35%
D. 37.5%
E. 75%
Show Spoiler
D.


Q9)What is the greatest value of n such that 30!/6^n is an integer?

A. 11
B. 12
C. 13
D. 14
E. 15
Show Spoiler
D.

Q10)The ratio of a to b is 4 to 5, where a and b are positive. If x equals a increased by 25 percent of a, and m equals b decreased by 20 percent of b, what is the value of m/x?

A. 2/5
B. 3/4
C. 4/5
D. 5/4
E. 3/2
Show Spoiler
C.


Q11)Ratio of two numbers x and y is 3:5. If x is increased by 20% and y is increased by 8 then the new ratio becomes 2:5. what is the ratio 2y:(x+6)

A. 2:5
B. 3:5
C. 4:5
D. 5:3
E. 10:3

Show Spoiler
D.


Q12)What is the greatest value of integer n such that 4^n is a factor of 26! ?

A. 6
B. 9
C. 10
D. 11
E. 12
Show Spoiler
D.

Q13)Each month, after Jill pays for rent, utilities, food, and other necessary expenses, she has one fifth of her net monthly salary left as discretionary income. Of this discretionary income, she puts 30% into a vacation fund, 20% into savings, and spends 35% on eating out and socializing. This leaves her with $96 dollar, which she typically uses for gifts and charitable causes. What is Jill’s net monthly salary?

(A) $2400
(B) $3200
(C) $6000
(D) $6400
(E) $960
Show Spoiler
B.


Q14)At a speed of 50 miles per hour, a certain car uses 1 gallon of gasoline every 30 miles. If the car starts with a full 12 gallon tank of gasoline and travels for 5 hours at 50 miles per hour, the amount of gasoline used would be what fraction of a full tank?

A. 3/25
B. 11/36
C. 7/12
D. 2/3
E. 25/36

Show Spoiler
E.


Q15)At a loading dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the day crew. If the night crew has 4/5 as many workers as the day crew, what fraction of all the boxes loaded by the two crews did the day crew load?

(A) 1/2
(B) 2/5
(C) 3/5
(D) 4/5
(E) 5/8

Show Spoiler
E.

Q16)At a certain school, the ratio of the number of second graders to the number of fourth graders is 8 to 5, and the ratio of the number of first graders to the number of second graders is 3 to 4. If the ratio of the number of third graders to the number of fourth graders is 3 to 2, what is the ratio of the number of first graders to the number of third graders?

A. 16 to 15
B. 9 to 5
C. 5 to 16
D. 5 to 4
E. 4 to 5

Show Spoiler
E.


Q17)If the ratio of the present age of Anna and Paula is 1 : 2, what could be the ratio of their respective ages 8 years ago??

a. 3 : 8
b. 4 : 7
c. 3 : 5
d. 2 : 3
e. 4 : 5
Show Spoiler
E.


Q18)Initially, the men and women in a room were in the ratio of 4 : 5. Then, 2 men entered the room and 3 women left the room. Then, the number of women doubled. Now there are 14 men in the room. How many women are currently in the room?

A. 12
B. 14
C. 15
D. 24
E. 36
Show Spoiler
D.


Q19)A feed store sells two varieties of birdseed: Brand A, which is 40% millet and 60% sunflower, and Brand B, which is 65% millet and 35% safflower. If a customer purchases a mix of the two types of birdseed that is 50% millet, what percent of the mix is Brand A?

A) 40%

B) 45%

C) 50 %

D) 60 %

E) 55 %
Show Spoiler
D.


Q20)Kumail, a noted sneaker enthusiast, has a collection consisting of sneakers made by Brand A and Brand B. 5/7 of the Brand A sneakers are low-tops, and 1/3 of the Brand A low-tops are running shoes. If Kumail owns 7 more pairs of Brand B sneakers than Brand A sneakers, what is the minimum possible number of pairs of sneakers in his collection?

A. 21
B. 28
C. 36
D. 40
E. 49
Show Spoiler
E.

Q21)If x dollars is invested at 10 percent for one year and y dollars is invested at 8 percent for one year, the annual income from the 10 percent investment will exceed the annual income from the 8 percent investment by $56. If $2,000 is the total amount invested, how much is invested at 8 percent?

a. $280
b. $800
c. $892
d. $1108
e. $1200
Show Spoiler
B.


Q22) Last year Elaine spent 20% of her annual earnings on rent. This year she earned 15% more than last year and she spent 30% of
her annual earnings on rent. The amount she spent on rent thisyear is what percent of the amount spent on rent last year?

(A) 152.5
(B) 164.5
(C) 167.5
(D) 172.5
(E) 177.5
Show Spoiler
D.


Q23)On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year. At the end of the year, during which he made no additional deposits or withdrawals, he had y dollars in the account. If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?

A. 2.04%
B. 6.12%
C. 8%
D. 8.25%
E. 10%
Show Spoiler
D.

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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Dec 2016, 01:59
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stonecold wrote:
Set X of has an average of 61. If the largest element is 7 greater than 6 times the smallest element, how many values out of {2,3,5,8,9,11,53,102,123,178,210,267,283,311,376,383,399,401} can be
a part of Set X?

(A) 18
(B) 13
(C) 9
(D) 7
(E) cannot be determined



Hi
Since the numbers of elements in set X is not given, we can play around with the total numbers and what is important is the smallest and largest values possible..

Smallest:-
Let the largest value be just above average..
So 6y+7>61....6y>61-7.....y>9

Largest:-
Let the smallest value be just below 61....
So 6*61+7>y.......373>y..

Range becomes 9<y<373..
Values in this range are 11,53,102,123,178,210,267,283,311.....9 values
Ans C
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  11 Dec 2016, 08:30
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Here is my take on this one ->

Let a be the smallest value.
Highest value in set =6a+7

As the values cannot be 50 each => a<61
and 6a+7>61 => a>9

Hence all values in the set must be greater than 9.
For a=61 => Largest term => 6*61+7=> 373

Hence all the values in set must lie in the range (9,373) exclusive.

Hence only 9 values out of the given values are possible.
Hene C


Similar Question -> a-set-of-numbers-has-an-average-of-50-if-the-largest-element-is-4-gre-203967.html
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 00:25
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 04:06
Column A --> 5^1, 5^2, 5^3, 5^4, 5^5 --> 5 values
Column B --> 2^0, 2^1, 2^2 --> 3 values

Number of factors that are divisible by 5 but not 3 = 5 * 3 = 15

Answer: B
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 04:10
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Vyshak wrote:
Column A --> 5^1, 5^2, 5^3, 5^4, 5^5 --> 5 values
Column B --> 2^0, 2^1, 2^2 --> 3 values

Number of factors that are divisible by 5 but not 3 = 5 * 3 = 15

Answer: B


Hi Vyshak, can you please elaborate this method if possible?

Thanks in advance.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 08:56
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Sirakri wrote:
Vyshak wrote:
Column A --> 5^1, 5^2, 5^3, 5^4, 5^5 --> 5 values
Column B --> 2^0, 2^1, 2^2 --> 3 values

Number of factors that are divisible by 5 but not 3 = 5 * 3 = 15

Answer: B


Hi Vyshak, can you please elaborate this method if possible?

Thanks in advance.


Hi Sirakri,

Its not any specific method. I just listed out the possible powers in two columns. Doing it will help us know that each value in column A will have 3 values in column B or vice versa.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 18:06
Here is my solution to this one =>
9919 => 10000-81=> 100^2-9^2=> 109*91 => 109*7*13 => Thus the greatest prime factor =109
NOTE=> 109 is a prime as it is not divisible by 2,3,5,7
In order to check whether a number is prime or not we just need to divide it by all the primes less than equal to the square root of that number.

Hence E.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  Updated on: 12 Jan 2017, 04:35
How many divisors does the positive integer \(N\) have?

A) \(27N^3\) has 16 factors.
B) \(90<N^3<200\)

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Originally posted by stonecold on 10 Jan 2017, 19:46.
Last edited by stonecold on 12 Jan 2017, 04:35, edited 2 times in total.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 20:56
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If t is a non negative integer, is integer k prime?
(1) k=13t + 13
(2) k=17! + 13


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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  10 Jan 2017, 22:52
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If N is a positive 3-digit number, is N prime?

(1)The hundreds digit of N is the sum of the tens and units digit. Tens and units digits are equal.
(2)N is odd.


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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  11 Jan 2017, 00:47
stonecold wrote:
If N is a positive 3-digit number, is N prime?

(1)The hundreds digit of K is the sum of the tens and units digit. Tens and units digits are equal.
(2)N is odd.


Source->Self-Made


Good one.

Let the number N = 100a + 10b + c, is N prime?

From St.1 :

a = b + c, and b=c ==> a = 2b,

a must be <= 9(since it is a digit). So possible values of a = 8, 6, 4, 2 and correspondig values of b and c= 4, 3, 2, 1

Possible values of N could be 844, 633, 422, and 211. So no unique answer. NOT sufficient.

From St.2:

N is odd. Clearly not sufficient.

By combining (1) and (2) N = 633 or 211. NOT unique.

Answer (E).

Thanks.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  11 Jan 2017, 02:34
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stonecold wrote:
If N is a positive 3-digit number, is N prime?

(1)The hundreds digit of K is the sum of the tens and units digit. Tens and units digits are equal.
(2)N is odd.


Source->Self-Made


stonecold
There is a typo in Statement 1 . you have mentioned K instead of N .

On the Solution part .
From Statement 1 , we know that N can be equal to 211, 422, 633, and 844 .
As a note: Only 211 is a prime here .
However , we don't get a unique value. So , Statement 1 is not Sufficient

Statement 2, Says N is odd
N can be 123 (Not prime) or 131 ( Prime ) .
No unique answer . So , Statement 2 is not Sufficient

Combine .
N can be equal to 211 or 633
No unique value .

Answer is E .
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  11 Jan 2017, 18:29
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stonecold wrote:
How many divisors does the positive integer \(N\) have?

A) \(27N^3\) has 16 factors.
B) \(90<N^3<200\)

Source->Self-Made



Hi....

Let's see the statements..

A) \(27N^3\) has 16 factors.
\(3^3*N^3\) has (3+1)(3+1)=16 factors...
The number of factors are always taken by getting the number into prime factors.
So here both 3 and N are PRIME numbers..
So N will have only two factors-- 1 and N..
Suff

B) \(90<N^3<200\)
Only N as 5 fits in..
4^3=64 and 6^3=216..
Sufficient

D
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post  12 Jan 2017, 03:05
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stonecold wrote:
How many divisors does the positive integer \(N\) have?

A) \(27N^3\) has 16 factors.
B) \(90<N^3<200\)

Source->Self-Made



From Statement 1 .
Given \(27N^3\) has 16 factors => \(3^3 N^3\) has 16 factors
Total factors are calculated by adding +1 to prime exponents and multiplying . In this case (3+1)(3+1) = 16
(3+1) is coming from 3 and another (3+1) need to come from N
N must be a prime factor and can't be a composite number because we will get more multipliers and factors will be greater than 16 .
As N is a prime number . We know it has only 2 factors .

Statement 1 is sufficient .

From Statement 2 .
Only 5^3 fits . N= 5
Statement 2 is sufficient .

Answer is D .
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